## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration.

#### Example 1

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 1)

A rocket weighing \(3000\) kg is traveling at a constant \(170\) meters per second. Then when \(t=57000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(6000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=66000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 50 \, u\left({t} - 57000\right) - 50 \, u\left({t} - 63000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{60} \, {\left({t} - 57000\right)} u\left({t} - 57000\right) - \frac{1}{60} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) + 170\]

It follows that when \(t=66000\), the velocity of the rocket is \(270\) meters per second.

#### Example 2

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 2)

A rocket weighing \(2300\) kg is traveling at a constant \(200\) meters per second. Then when \(t=13800\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(6900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=25300\).

#### Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 20 \, u\left({t} - 13800\right) - 20 \, u\left({t} - 20700\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{1}{115} \, {\left({t} - 13800\right)} u\left({t} - 13800\right) - \frac{1}{115} \, {\left({t} - 20700\right)} u\left({t} - 20700\right) + 200\]

It follows that when \(t=25300\), the velocity of the rocket is \(260\) meters per second.

#### Example 3

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 3)

A rocket weighing \(4200\) kg is traveling at a constant \(90\) meters per second. Then when \(t=67200\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(12600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=88200\).

#### Answer.

An IVP modeling this scenario is given by:

\[4200 \, {v'} = 90 \, u\left({t} - 67200\right) - 90 \, u\left({t} - 79800\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{3}{140} \, {\left({t} - 67200\right)} u\left({t} - 67200\right) - \frac{3}{140} \, {\left({t} - 79800\right)} u\left({t} - 79800\right) + 90\]

It follows that when \(t=88200\), the velocity of the rocket is \(360\) meters per second.

#### Example 4

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 4)

A rocket weighing \(1600\) kg is traveling at a constant \(130\) meters per second. Then when \(t=28800\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=30400\).

#### Answer.

An IVP modeling this scenario is given by:

\[1600 \, {v'} = 20 \, u\left({t} - 28800\right) - 20 \, u\left({t} - 33600\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{1}{80} \, {\left({t} - 28800\right)} u\left({t} - 28800\right) - \frac{1}{80} \, {\left({t} - 33600\right)} u\left({t} - 33600\right) + 130\]

It follows that when \(t=30400\), the velocity of the rocket is \(150\) meters per second.

#### Example 5

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 5)

A rocket weighing \(2800\) kg is traveling at a constant \(10\) meters per second. Then when \(t=28000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(8400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=33600\).

#### Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 30 \, u\left({t} - 28000\right) - 30 \, u\left({t} - 36400\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{3}{280} \, {\left({t} - 28000\right)} u\left({t} - 28000\right) - \frac{3}{280} \, {\left({t} - 36400\right)} u\left({t} - 36400\right) + 10\]

It follows that when \(t=33600\), the velocity of the rocket is \(70\) meters per second.

#### Example 6

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 6)

A rocket weighing \(4500\) kg is traveling at a constant \(180\) meters per second. Then when \(t=63000\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(9000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=72000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 40 \, u\left({t} - 63000\right) - 40 \, u\left({t} - 72000\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{2}{225} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) - \frac{2}{225} \, {\left({t} - 72000\right)} u\left({t} - 72000\right) + 180\]

It follows that when \(t=72000\), the velocity of the rocket is \(260\) meters per second.

#### Example 7

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 7)

A rocket weighing \(3900\) kg is traveling at a constant \(170\) meters per second. Then when \(t=62400\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=81900\).

#### Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 60 \, u\left({t} - 62400\right) - 60 \, u\left({t} - 78000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{65} \, {\left({t} - 62400\right)} u\left({t} - 62400\right) - \frac{1}{65} \, {\left({t} - 78000\right)} u\left({t} - 78000\right) + 170\]

It follows that when \(t=81900\), the velocity of the rocket is \(410\) meters per second.

#### Example 8

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 8)

A rocket weighing \(4600\) kg is traveling at a constant \(180\) meters per second. Then when \(t=92000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=110400\).

#### Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 60 \, u\left({t} - 92000\right) - 60 \, u\left({t} - 101200\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{3}{230} \, {\left({t} - 92000\right)} u\left({t} - 92000\right) - \frac{3}{230} \, {\left({t} - 101200\right)} u\left({t} - 101200\right) + 180\]

It follows that when \(t=110400\), the velocity of the rocket is \(300\) meters per second.

#### Example 9

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 9)

A rocket weighing \(1500\) kg is traveling at a constant \(130\) meters per second. Then when \(t=6000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9000\).

#### Answer.

An IVP modeling this scenario is given by:

\[1500 \, {v'} = 60 \, u\left({t} - 6000\right) - 60 \, u\left({t} - 9000\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{1}{25} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) - \frac{1}{25} \, {\left({t} - 9000\right)} u\left({t} - 9000\right) + 130\]

It follows that when \(t=9000\), the velocity of the rocket is \(250\) meters per second.

#### Example 10

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 10)

A rocket weighing \(2100\) kg is traveling at a constant \(200\) meters per second. Then when \(t=39900\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(4200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=42000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2100 \, {v'} = 50 \, u\left({t} - 39900\right) - 50 \, u\left({t} - 44100\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{1}{42} \, {\left({t} - 39900\right)} u\left({t} - 39900\right) - \frac{1}{42} \, {\left({t} - 44100\right)} u\left({t} - 44100\right) + 200\]

It follows that when \(t=42000\), the velocity of the rocket is \(250\) meters per second.

#### Example 11

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 11)

A rocket weighing \(2400\) kg is traveling at a constant \(100\) meters per second. Then when \(t=12000\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14400\).

#### Answer.

An IVP modeling this scenario is given by:

\[2400 \, {v'} = 20 \, u\left({t} - 12000\right) - 20 \, u\left({t} - 16800\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{120} \, {\left({t} - 12000\right)} u\left({t} - 12000\right) - \frac{1}{120} \, {\left({t} - 16800\right)} u\left({t} - 16800\right) + 100\]

It follows that when \(t=14400\), the velocity of the rocket is \(120\) meters per second.

#### Example 12

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 12)

A rocket weighing \(3000\) kg is traveling at a constant \(100\) meters per second. Then when \(t=24000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=48000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 50 \, u\left({t} - 24000\right) - 50 \, u\left({t} - 36000\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{60} \, {\left({t} - 24000\right)} u\left({t} - 24000\right) - \frac{1}{60} \, {\left({t} - 36000\right)} u\left({t} - 36000\right) + 100\]

It follows that when \(t=48000\), the velocity of the rocket is \(300\) meters per second.

#### Example 13

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 13)

A rocket weighing \(4500\) kg is traveling at a constant \(50\) meters per second. Then when \(t=13500\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(18000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=27000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 20 \, u\left({t} - 13500\right) - 20 \, u\left({t} - 31500\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{225} \, {\left({t} - 13500\right)} u\left({t} - 13500\right) - \frac{1}{225} \, {\left({t} - 31500\right)} u\left({t} - 31500\right) + 50\]

It follows that when \(t=27000\), the velocity of the rocket is \(110\) meters per second.

#### Example 14

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 14)

A rocket weighing \(700\) kg is traveling at a constant \(10\) meters per second. Then when \(t=6300\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(1400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=8400\).

#### Answer.

An IVP modeling this scenario is given by:

\[700 \, {v'} = 90 \, u\left({t} - 6300\right) - 90 \, u\left({t} - 7700\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{9}{70} \, {\left({t} - 6300\right)} u\left({t} - 6300\right) - \frac{9}{70} \, {\left({t} - 7700\right)} u\left({t} - 7700\right) + 10\]

It follows that when \(t=8400\), the velocity of the rocket is \(190\) meters per second.

#### Example 15

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 15)

A rocket weighing \(3400\) kg is traveling at a constant \(140\) meters per second. Then when \(t=44200\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(13600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=51000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3400 \, {v'} = 20 \, u\left({t} - 44200\right) - 20 \, u\left({t} - 57800\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{1}{170} \, {\left({t} - 44200\right)} u\left({t} - 44200\right) - \frac{1}{170} \, {\left({t} - 57800\right)} u\left({t} - 57800\right) + 140\]

It follows that when \(t=51000\), the velocity of the rocket is \(180\) meters per second.

#### Example 16

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 16)

A rocket weighing \(2800\) kg is traveling at a constant \(90\) meters per second. Then when \(t=0\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(11200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=19600\).

#### Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = -10 \, u\left({t} - 11200\right) + 10 \, u\left({t}\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = -\frac{1}{280} \, {\left({t} - 11200\right)} u\left({t} - 11200\right) + \frac{1}{280} \, {t} u\left({t}\right) + 90\]

It follows that when \(t=19600\), the velocity of the rocket is \(130\) meters per second.

#### Example 17

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 17)

A rocket weighing \(2300\) kg is traveling at a constant \(100\) meters per second. Then when \(t=18400\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=32200\).

#### Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 50 \, u\left({t} - 18400\right) - 50 \, u\left({t} - 27600\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{46} \, {\left({t} - 18400\right)} u\left({t} - 18400\right) - \frac{1}{46} \, {\left({t} - 27600\right)} u\left({t} - 27600\right) + 100\]

It follows that when \(t=32200\), the velocity of the rocket is \(300\) meters per second.

#### Example 18

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 18)

A rocket weighing \(1900\) kg is traveling at a constant \(200\) meters per second. Then when \(t=38000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=41800\).

#### Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 60 \, u\left({t} - 38000\right) - 60 \, u\left({t} - 41800\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{3}{95} \, {\left({t} - 38000\right)} u\left({t} - 38000\right) - \frac{3}{95} \, {\left({t} - 41800\right)} u\left({t} - 41800\right) + 200\]

It follows that when \(t=41800\), the velocity of the rocket is \(320\) meters per second.

#### Example 19

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 19)

A rocket weighing \(4900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=24500\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(14700\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=34300\).

#### Answer.

An IVP modeling this scenario is given by:

\[4900 \, {v'} = 100 \, u\left({t} - 24500\right) - 100 \, u\left({t} - 39200\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{49} \, {\left({t} - 24500\right)} u\left({t} - 24500\right) - \frac{1}{49} \, {\left({t} - 39200\right)} u\left({t} - 39200\right) + 80\]

It follows that when \(t=34300\), the velocity of the rocket is \(280\) meters per second.

#### Example 20

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 20)

A rocket weighing \(3400\) kg is traveling at a constant \(90\) meters per second. Then when \(t=57800\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(13600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=61200\).

#### Answer.

An IVP modeling this scenario is given by:

\[3400 \, {v'} = 60 \, u\left({t} - 57800\right) - 60 \, u\left({t} - 71400\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{3}{170} \, {\left({t} - 57800\right)} u\left({t} - 57800\right) - \frac{3}{170} \, {\left({t} - 71400\right)} u\left({t} - 71400\right) + 90\]

It follows that when \(t=61200\), the velocity of the rocket is \(150\) meters per second.

#### Example 21

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 21)

A rocket weighing \(100\) kg is traveling at a constant \(50\) meters per second. Then when \(t=1900\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2100\).

#### Answer.

An IVP modeling this scenario is given by:

\[100 \, {v'} = 30 \, u\left({t} - 1900\right) - 30 \, u\left({t} - 2200\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{3}{10} \, {\left({t} - 1900\right)} u\left({t} - 1900\right) - \frac{3}{10} \, {\left({t} - 2200\right)} u\left({t} - 2200\right) + 50\]

It follows that when \(t=2100\), the velocity of the rocket is \(110\) meters per second.

#### Example 22

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 22)

A rocket weighing \(2300\) kg is traveling at a constant \(60\) meters per second. Then when \(t=43700\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(4600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=52900\).

#### Answer.

An IVP modeling this scenario is given by:

\[2300 \, {v'} = 20 \, u\left({t} - 43700\right) - 20 \, u\left({t} - 48300\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{115} \, {\left({t} - 43700\right)} u\left({t} - 43700\right) - \frac{1}{115} \, {\left({t} - 48300\right)} u\left({t} - 48300\right) + 60\]

It follows that when \(t=52900\), the velocity of the rocket is \(100\) meters per second.

#### Example 23

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 23)

A rocket weighing \(800\) kg is traveling at a constant \(20\) meters per second. Then when \(t=14400\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(2400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15200\).

#### Answer.

An IVP modeling this scenario is given by:

\[800 \, {v'} = 20 \, u\left({t} - 14400\right) - 20 \, u\left({t} - 16800\right)\hspace{2em}v(0)= 20\]

This IVP solves to:

\[{v} = \frac{1}{40} \, {\left({t} - 14400\right)} u\left({t} - 14400\right) - \frac{1}{40} \, {\left({t} - 16800\right)} u\left({t} - 16800\right) + 20\]

It follows that when \(t=15200\), the velocity of the rocket is \(40\) meters per second.

#### Example 24

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 24)

A rocket weighing \(4500\) kg is traveling at a constant \(90\) meters per second. Then when \(t=76500\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(13500\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=90000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 40 \, u\left({t} - 76500\right) - 40 \, u\left({t} - 90000\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{2}{225} \, {\left({t} - 76500\right)} u\left({t} - 76500\right) - \frac{2}{225} \, {\left({t} - 90000\right)} u\left({t} - 90000\right) + 90\]

It follows that when \(t=90000\), the velocity of the rocket is \(210\) meters per second.

#### Example 25

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 25)

A rocket weighing \(3900\) kg is traveling at a constant \(100\) meters per second. Then when \(t=27300\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=58500\).

#### Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 100 \, u\left({t} - 27300\right) - 100 \, u\left({t} - 42900\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{39} \, {\left({t} - 27300\right)} u\left({t} - 27300\right) - \frac{1}{39} \, {\left({t} - 42900\right)} u\left({t} - 42900\right) + 100\]

It follows that when \(t=58500\), the velocity of the rocket is \(500\) meters per second.

#### Example 26

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 26)

A rocket weighing \(3800\) kg is traveling at a constant \(190\) meters per second. Then when \(t=41800\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(15200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=72200\).

#### Answer.

An IVP modeling this scenario is given by:

\[3800 \, {v'} = 10 \, u\left({t} - 41800\right) - 10 \, u\left({t} - 57000\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{1}{380} \, {\left({t} - 41800\right)} u\left({t} - 41800\right) - \frac{1}{380} \, {\left({t} - 57000\right)} u\left({t} - 57000\right) + 190\]

It follows that when \(t=72200\), the velocity of the rocket is \(230\) meters per second.

#### Example 27

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 27)

A rocket weighing \(4100\) kg is traveling at a constant \(40\) meters per second. Then when \(t=4100\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(8200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=16400\).

#### Answer.

An IVP modeling this scenario is given by:

\[4100 \, {v'} = 100 \, u\left({t} - 4100\right) - 100 \, u\left({t} - 12300\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{1}{41} \, {\left({t} - 4100\right)} u\left({t} - 4100\right) - \frac{1}{41} \, {\left({t} - 12300\right)} u\left({t} - 12300\right) + 40\]

It follows that when \(t=16400\), the velocity of the rocket is \(240\) meters per second.

#### Example 28

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 28)

A rocket weighing \(4900\) kg is traveling at a constant \(90\) meters per second. Then when \(t=19600\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(9800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=39200\).

#### Answer.

An IVP modeling this scenario is given by:

\[4900 \, {v'} = 90 \, u\left({t} - 19600\right) - 90 \, u\left({t} - 29400\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{9}{490} \, {\left({t} - 19600\right)} u\left({t} - 19600\right) - \frac{9}{490} \, {\left({t} - 29400\right)} u\left({t} - 29400\right) + 90\]

It follows that when \(t=39200\), the velocity of the rocket is \(270\) meters per second.

#### Example 29

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 29)

A rocket weighing \(2800\) kg is traveling at a constant \(170\) meters per second. Then when \(t=30800\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(5600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=39200\).

#### Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 40 \, u\left({t} - 30800\right) - 40 \, u\left({t} - 36400\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{70} \, {\left({t} - 30800\right)} u\left({t} - 30800\right) - \frac{1}{70} \, {\left({t} - 36400\right)} u\left({t} - 36400\right) + 170\]

It follows that when \(t=39200\), the velocity of the rocket is \(250\) meters per second.

#### Example 30

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 30)

A rocket weighing \(3900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=54600\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(15600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=58500\).

#### Answer.

An IVP modeling this scenario is given by:

\[3900 \, {v'} = 10 \, u\left({t} - 54600\right) - 10 \, u\left({t} - 70200\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{390} \, {\left({t} - 54600\right)} u\left({t} - 54600\right) - \frac{1}{390} \, {\left({t} - 70200\right)} u\left({t} - 70200\right) + 80\]

It follows that when \(t=58500\), the velocity of the rocket is \(90\) meters per second.

#### Example 31

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 31)

A rocket weighing \(3200\) kg is traveling at a constant \(10\) meters per second. Then when \(t=57600\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(6400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=64000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3200 \, {v'} = 70 \, u\left({t} - 57600\right) - 70 \, u\left({t} - 64000\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{7}{320} \, {\left({t} - 57600\right)} u\left({t} - 57600\right) - \frac{7}{320} \, {\left({t} - 64000\right)} u\left({t} - 64000\right) + 10\]

It follows that when \(t=64000\), the velocity of the rocket is \(150\) meters per second.

#### Example 32

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 32)

A rocket weighing \(2800\) kg is traveling at a constant \(110\) meters per second. Then when \(t=14000\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(11200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=36400\).

#### Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 80 \, u\left({t} - 14000\right) - 80 \, u\left({t} - 25200\right)\hspace{2em}v(0)= 110\]

This IVP solves to:

\[{v} = \frac{1}{35} \, {\left({t} - 14000\right)} u\left({t} - 14000\right) - \frac{1}{35} \, {\left({t} - 25200\right)} u\left({t} - 25200\right) + 110\]

It follows that when \(t=36400\), the velocity of the rocket is \(430\) meters per second.

#### Example 33

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 33)

A rocket weighing \(2100\) kg is traveling at a constant \(10\) meters per second. Then when \(t=2100\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(8400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14700\).

#### Answer.

An IVP modeling this scenario is given by:

\[2100 \, {v'} = 90 \, u\left({t} - 2100\right) - 90 \, u\left({t} - 10500\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{3}{70} \, {\left({t} - 2100\right)} u\left({t} - 2100\right) - \frac{3}{70} \, {\left({t} - 10500\right)} u\left({t} - 10500\right) + 10\]

It follows that when \(t=14700\), the velocity of the rocket is \(370\) meters per second.

#### Example 34

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 34)

A rocket weighing \(4300\) kg is traveling at a constant \(130\) meters per second. Then when \(t=77400\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(12900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=103200\).

#### Answer.

An IVP modeling this scenario is given by:

\[4300 \, {v'} = 40 \, u\left({t} - 77400\right) - 40 \, u\left({t} - 90300\right)\hspace{2em}v(0)= 130\]

This IVP solves to:

\[{v} = \frac{2}{215} \, {\left({t} - 77400\right)} u\left({t} - 77400\right) - \frac{2}{215} \, {\left({t} - 90300\right)} u\left({t} - 90300\right) + 130\]

It follows that when \(t=103200\), the velocity of the rocket is \(250\) meters per second.

#### Example 35

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 35)

A rocket weighing \(4500\) kg is traveling at a constant \(60\) meters per second. Then when \(t=63000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(18000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=99000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 60 \, u\left({t} - 63000\right) - 60 \, u\left({t} - 81000\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{75} \, {\left({t} - 63000\right)} u\left({t} - 63000\right) - \frac{1}{75} \, {\left({t} - 81000\right)} u\left({t} - 81000\right) + 60\]

It follows that when \(t=99000\), the velocity of the rocket is \(300\) meters per second.

#### Example 36

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 36)

A rocket weighing \(1900\) kg is traveling at a constant \(170\) meters per second. Then when \(t=13300\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15200\).

#### Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 60 \, u\left({t} - 13300\right) - 60 \, u\left({t} - 17100\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{3}{95} \, {\left({t} - 13300\right)} u\left({t} - 13300\right) - \frac{3}{95} \, {\left({t} - 17100\right)} u\left({t} - 17100\right) + 170\]

It follows that when \(t=15200\), the velocity of the rocket is \(230\) meters per second.

#### Example 37

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 37)

A rocket weighing \(3600\) kg is traveling at a constant \(160\) meters per second. Then when \(t=25200\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(7200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=36000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3600 \, {v'} = 100 \, u\left({t} - 25200\right) - 100 \, u\left({t} - 32400\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{1}{36} \, {\left({t} - 25200\right)} u\left({t} - 25200\right) - \frac{1}{36} \, {\left({t} - 32400\right)} u\left({t} - 32400\right) + 160\]

It follows that when \(t=36000\), the velocity of the rocket is \(360\) meters per second.

#### Example 38

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 38)

A rocket weighing \(1000\) kg is traveling at a constant \(50\) meters per second. Then when \(t=1000\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(4000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9000\).

#### Answer.

An IVP modeling this scenario is given by:

\[1000 \, {v'} = 10 \, u\left({t} - 1000\right) - 10 \, u\left({t} - 5000\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{100} \, {\left({t} - 1000\right)} u\left({t} - 1000\right) - \frac{1}{100} \, {\left({t} - 5000\right)} u\left({t} - 5000\right) + 50\]

It follows that when \(t=9000\), the velocity of the rocket is \(90\) meters per second.

#### Example 39

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 39)

A rocket weighing \(1100\) kg is traveling at a constant \(170\) meters per second. Then when \(t=0\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(3300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2200\).

#### Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = -70 \, u\left({t} - 3300\right) + 70 \, u\left({t}\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = -\frac{7}{110} \, {\left({t} - 3300\right)} u\left({t} - 3300\right) + \frac{7}{110} \, {t} u\left({t}\right) + 170\]

It follows that when \(t=2200\), the velocity of the rocket is \(310\) meters per second.

#### Example 40

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 40)

A rocket weighing \(1900\) kg is traveling at a constant \(80\) meters per second. Then when \(t=19000\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(3800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=24700\).

#### Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 20 \, u\left({t} - 19000\right) - 20 \, u\left({t} - 22800\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{1}{95} \, {\left({t} - 19000\right)} u\left({t} - 19000\right) - \frac{1}{95} \, {\left({t} - 22800\right)} u\left({t} - 22800\right) + 80\]

It follows that when \(t=24700\), the velocity of the rocket is \(120\) meters per second.

#### Example 41

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 41)

A rocket weighing \(5000\) kg is traveling at a constant \(70\) meters per second. Then when \(t=50000\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(10000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=70000\).

#### Answer.

An IVP modeling this scenario is given by:

\[5000 \, {v'} = 100 \, u\left({t} - 50000\right) - 100 \, u\left({t} - 60000\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{1}{50} \, {\left({t} - 50000\right)} u\left({t} - 50000\right) - \frac{1}{50} \, {\left({t} - 60000\right)} u\left({t} - 60000\right) + 70\]

It follows that when \(t=70000\), the velocity of the rocket is \(270\) meters per second.

#### Example 42

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 42)

A rocket weighing \(4600\) kg is traveling at a constant \(80\) meters per second. Then when \(t=32200\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(18400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=50600\).

#### Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 80 \, u\left({t} - 32200\right) - 80 \, u\left({t} - 50600\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{2}{115} \, {\left({t} - 32200\right)} u\left({t} - 32200\right) - \frac{2}{115} \, {\left({t} - 50600\right)} u\left({t} - 50600\right) + 80\]

It follows that when \(t=50600\), the velocity of the rocket is \(400\) meters per second.

#### Example 43

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 43)

A rocket weighing \(3600\) kg is traveling at a constant \(50\) meters per second. Then when \(t=39600\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(14400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=64800\).

#### Answer.

An IVP modeling this scenario is given by:

\[3600 \, {v'} = 50 \, u\left({t} - 39600\right) - 50 \, u\left({t} - 54000\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{72} \, {\left({t} - 39600\right)} u\left({t} - 39600\right) - \frac{1}{72} \, {\left({t} - 54000\right)} u\left({t} - 54000\right) + 50\]

It follows that when \(t=64800\), the velocity of the rocket is \(250\) meters per second.

#### Example 44

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 44)

A rocket weighing \(200\) kg is traveling at a constant \(10\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=2800\).

#### Answer.

An IVP modeling this scenario is given by:

\[200 \, {v'} = 10 \, u\left({t} - 2400\right) - 10 \, u\left({t} - 2800\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{1}{20} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{20} \, {\left({t} - 2800\right)} u\left({t} - 2800\right) + 10\]

It follows that when \(t=2800\), the velocity of the rocket is \(30\) meters per second.

#### Example 45

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 45)

A rocket weighing \(3000\) kg is traveling at a constant \(170\) meters per second. Then when \(t=6000\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=15000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = 10 \, u\left({t} - 6000\right) - 10 \, u\left({t} - 18000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{300} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) - \frac{1}{300} \, {\left({t} - 18000\right)} u\left({t} - 18000\right) + 170\]

It follows that when \(t=15000\), the velocity of the rocket is \(200\) meters per second.

#### Example 46

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 46)

A rocket weighing \(4000\) kg is traveling at a constant \(70\) meters per second. Then when \(t=76000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(12000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=80000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4000 \, {v'} = 50 \, u\left({t} - 76000\right) - 50 \, u\left({t} - 88000\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{1}{80} \, {\left({t} - 76000\right)} u\left({t} - 76000\right) - \frac{1}{80} \, {\left({t} - 88000\right)} u\left({t} - 88000\right) + 70\]

It follows that when \(t=80000\), the velocity of the rocket is \(120\) meters per second.

#### Example 47

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 47)

A rocket weighing \(1800\) kg is traveling at a constant \(170\) meters per second. Then when \(t=1800\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(3600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=5400\).

#### Answer.

An IVP modeling this scenario is given by:

\[1800 \, {v'} = 10 \, u\left({t} - 1800\right) - 10 \, u\left({t} - 5400\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{180} \, {\left({t} - 1800\right)} u\left({t} - 1800\right) - \frac{1}{180} \, {\left({t} - 5400\right)} u\left({t} - 5400\right) + 170\]

It follows that when \(t=5400\), the velocity of the rocket is \(190\) meters per second.

#### Example 48

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 48)

A rocket weighing \(2600\) kg is traveling at a constant \(40\) meters per second. Then when \(t=39000\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(10400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=52000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2600 \, {v'} = 70 \, u\left({t} - 39000\right) - 70 \, u\left({t} - 49400\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{7}{260} \, {\left({t} - 39000\right)} u\left({t} - 39000\right) - \frac{7}{260} \, {\left({t} - 49400\right)} u\left({t} - 49400\right) + 40\]

It follows that when \(t=52000\), the velocity of the rocket is \(320\) meters per second.

#### Example 49

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 49)

A rocket weighing \(1200\) kg is traveling at a constant \(100\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(3600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9600\).

#### Answer.

An IVP modeling this scenario is given by:

\[1200 \, {v'} = 30 \, u\left({t} - 2400\right) - 30 \, u\left({t} - 6000\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{1}{40} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{40} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) + 100\]

It follows that when \(t=9600\), the velocity of the rocket is \(190\) meters per second.

#### Example 50

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 50)

A rocket weighing \(300\) kg is traveling at a constant \(120\) meters per second. Then when \(t=2400\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=3000\).

#### Answer.

An IVP modeling this scenario is given by:

\[300 \, {v'} = 100 \, u\left({t} - 2400\right) - 100 \, u\left({t} - 3300\right)\hspace{2em}v(0)= 120\]

This IVP solves to:

\[{v} = \frac{1}{3} \, {\left({t} - 2400\right)} u\left({t} - 2400\right) - \frac{1}{3} \, {\left({t} - 3300\right)} u\left({t} - 3300\right) + 120\]

It follows that when \(t=3000\), the velocity of the rocket is \(320\) meters per second.

#### Example 51

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 51)

A rocket weighing \(1500\) kg is traveling at a constant \(60\) meters per second. Then when \(t=24000\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(4500\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=33000\).

#### Answer.

An IVP modeling this scenario is given by:

\[1500 \, {v'} = 70 \, u\left({t} - 24000\right) - 70 \, u\left({t} - 28500\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{7}{150} \, {\left({t} - 24000\right)} u\left({t} - 24000\right) - \frac{7}{150} \, {\left({t} - 28500\right)} u\left({t} - 28500\right) + 60\]

It follows that when \(t=33000\), the velocity of the rocket is \(270\) meters per second.

#### Example 52

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 52)

A rocket weighing \(4500\) kg is traveling at a constant \(70\) meters per second. Then when \(t=49500\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(18000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=72000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 60 \, u\left({t} - 49500\right) - 60 \, u\left({t} - 67500\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{1}{75} \, {\left({t} - 49500\right)} u\left({t} - 49500\right) - \frac{1}{75} \, {\left({t} - 67500\right)} u\left({t} - 67500\right) + 70\]

It follows that when \(t=72000\), the velocity of the rocket is \(310\) meters per second.

#### Example 53

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 53)

A rocket weighing \(2900\) kg is traveling at a constant \(150\) meters per second. Then when \(t=37700\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(5800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=40600\).

#### Answer.

An IVP modeling this scenario is given by:

\[2900 \, {v'} = 90 \, u\left({t} - 37700\right) - 90 \, u\left({t} - 43500\right)\hspace{2em}v(0)= 150\]

This IVP solves to:

\[{v} = \frac{9}{290} \, {\left({t} - 37700\right)} u\left({t} - 37700\right) - \frac{9}{290} \, {\left({t} - 43500\right)} u\left({t} - 43500\right) + 150\]

It follows that when \(t=40600\), the velocity of the rocket is \(240\) meters per second.

#### Example 54

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 54)

A rocket weighing \(3300\) kg is traveling at a constant \(140\) meters per second. Then when \(t=56100\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(13200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=62700\).

#### Answer.

An IVP modeling this scenario is given by:

\[3300 \, {v'} = 60 \, u\left({t} - 56100\right) - 60 \, u\left({t} - 69300\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{1}{55} \, {\left({t} - 56100\right)} u\left({t} - 56100\right) - \frac{1}{55} \, {\left({t} - 69300\right)} u\left({t} - 69300\right) + 140\]

It follows that when \(t=62700\), the velocity of the rocket is \(260\) meters per second.

#### Example 55

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 55)

A rocket weighing \(4600\) kg is traveling at a constant \(200\) meters per second. Then when \(t=59800\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=69000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 20 \, u\left({t} - 59800\right) - 20 \, u\left({t} - 69000\right)\hspace{2em}v(0)= 200\]

This IVP solves to:

\[{v} = \frac{1}{230} \, {\left({t} - 59800\right)} u\left({t} - 59800\right) - \frac{1}{230} \, {\left({t} - 69000\right)} u\left({t} - 69000\right) + 200\]

It follows that when \(t=69000\), the velocity of the rocket is \(240\) meters per second.

#### Example 56

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 56)

A rocket weighing \(1100\) kg is traveling at a constant \(120\) meters per second. Then when \(t=6600\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(3300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=12100\).

#### Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = 80 \, u\left({t} - 6600\right) - 80 \, u\left({t} - 9900\right)\hspace{2em}v(0)= 120\]

This IVP solves to:

\[{v} = \frac{4}{55} \, {\left({t} - 6600\right)} u\left({t} - 6600\right) - \frac{4}{55} \, {\left({t} - 9900\right)} u\left({t} - 9900\right) + 120\]

It follows that when \(t=12100\), the velocity of the rocket is \(360\) meters per second.

#### Example 57

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 57)

A rocket weighing \(1100\) kg is traveling at a constant \(150\) meters per second. Then when \(t=13200\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(4400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=19800\).

#### Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = 80 \, u\left({t} - 13200\right) - 80 \, u\left({t} - 17600\right)\hspace{2em}v(0)= 150\]

This IVP solves to:

\[{v} = \frac{4}{55} \, {\left({t} - 13200\right)} u\left({t} - 13200\right) - \frac{4}{55} \, {\left({t} - 17600\right)} u\left({t} - 17600\right) + 150\]

It follows that when \(t=19800\), the velocity of the rocket is \(470\) meters per second.

#### Example 58

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 58)

A rocket weighing \(2000\) kg is traveling at a constant \(140\) meters per second. Then when \(t=12000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(4000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2000 \, {v'} = 30 \, u\left({t} - 12000\right) - 30 \, u\left({t} - 16000\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{3}{200} \, {\left({t} - 12000\right)} u\left({t} - 12000\right) - \frac{3}{200} \, {\left({t} - 16000\right)} u\left({t} - 16000\right) + 140\]

It follows that when \(t=14000\), the velocity of the rocket is \(170\) meters per second.

#### Example 59

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 59)

A rocket weighing \(200\) kg is traveling at a constant \(50\) meters per second. Then when \(t=3200\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=3600\).

#### Answer.

An IVP modeling this scenario is given by:

\[200 \, {v'} = 80 \, u\left({t} - 3200\right) - 80 \, u\left({t} - 3800\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{2}{5} \, {\left({t} - 3200\right)} u\left({t} - 3200\right) - \frac{2}{5} \, {\left({t} - 3800\right)} u\left({t} - 3800\right) + 50\]

It follows that when \(t=3600\), the velocity of the rocket is \(210\) meters per second.

#### Example 60

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 60)

A rocket weighing \(1700\) kg is traveling at a constant \(60\) meters per second. Then when \(t=20400\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(3400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=27200\).

#### Answer.

An IVP modeling this scenario is given by:

\[1700 \, {v'} = 10 \, u\left({t} - 20400\right) - 10 \, u\left({t} - 23800\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{170} \, {\left({t} - 20400\right)} u\left({t} - 20400\right) - \frac{1}{170} \, {\left({t} - 23800\right)} u\left({t} - 23800\right) + 60\]

It follows that when \(t=27200\), the velocity of the rocket is \(80\) meters per second.

#### Example 61

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 61)

A rocket weighing \(5000\) kg is traveling at a constant \(30\) meters per second. Then when \(t=75000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(20000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=100000\).

#### Answer.

An IVP modeling this scenario is given by:

\[5000 \, {v'} = 30 \, u\left({t} - 75000\right) - 30 \, u\left({t} - 95000\right)\hspace{2em}v(0)= 30\]

This IVP solves to:

\[{v} = \frac{3}{500} \, {\left({t} - 75000\right)} u\left({t} - 75000\right) - \frac{3}{500} \, {\left({t} - 95000\right)} u\left({t} - 95000\right) + 30\]

It follows that when \(t=100000\), the velocity of the rocket is \(150\) meters per second.

#### Example 62

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 62)

A rocket weighing \(2000\) kg is traveling at a constant \(10\) meters per second. Then when \(t=12000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(6000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=16000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2000 \, {v'} = 30 \, u\left({t} - 12000\right) - 30 \, u\left({t} - 18000\right)\hspace{2em}v(0)= 10\]

This IVP solves to:

\[{v} = \frac{3}{200} \, {\left({t} - 12000\right)} u\left({t} - 12000\right) - \frac{3}{200} \, {\left({t} - 18000\right)} u\left({t} - 18000\right) + 10\]

It follows that when \(t=16000\), the velocity of the rocket is \(70\) meters per second.

#### Example 63

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 63)

A rocket weighing \(600\) kg is traveling at a constant \(170\) meters per second. Then when \(t=3600\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(1800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=4800\).

#### Answer.

An IVP modeling this scenario is given by:

\[600 \, {v'} = 30 \, u\left({t} - 3600\right) - 30 \, u\left({t} - 5400\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{20} \, {\left({t} - 3600\right)} u\left({t} - 3600\right) - \frac{1}{20} \, {\left({t} - 5400\right)} u\left({t} - 5400\right) + 170\]

It follows that when \(t=4800\), the velocity of the rocket is \(230\) meters per second.

#### Example 64

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 64)

A rocket weighing \(3400\) kg is traveling at a constant \(190\) meters per second. Then when \(t=27200\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(13600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=47600\).

#### Answer.

An IVP modeling this scenario is given by:

\[3400 \, {v'} = 10 \, u\left({t} - 27200\right) - 10 \, u\left({t} - 40800\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{1}{340} \, {\left({t} - 27200\right)} u\left({t} - 27200\right) - \frac{1}{340} \, {\left({t} - 40800\right)} u\left({t} - 40800\right) + 190\]

It follows that when \(t=47600\), the velocity of the rocket is \(230\) meters per second.

#### Example 65

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 65)

A rocket weighing \(1600\) kg is traveling at a constant \(160\) meters per second. Then when \(t=32000\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=40000\).

#### Answer.

An IVP modeling this scenario is given by:

\[1600 \, {v'} = 60 \, u\left({t} - 32000\right) - 60 \, u\left({t} - 36800\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{3}{80} \, {\left({t} - 32000\right)} u\left({t} - 32000\right) - \frac{3}{80} \, {\left({t} - 36800\right)} u\left({t} - 36800\right) + 160\]

It follows that when \(t=40000\), the velocity of the rocket is \(340\) meters per second.

#### Example 66

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 66)

A rocket weighing \(900\) kg is traveling at a constant \(90\) meters per second. Then when \(t=1800\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(1800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=3600\).

#### Answer.

An IVP modeling this scenario is given by:

\[900 \, {v'} = 100 \, u\left({t} - 1800\right) - 100 \, u\left({t} - 3600\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{1}{9} \, {\left({t} - 1800\right)} u\left({t} - 1800\right) - \frac{1}{9} \, {\left({t} - 3600\right)} u\left({t} - 3600\right) + 90\]

It follows that when \(t=3600\), the velocity of the rocket is \(290\) meters per second.

#### Example 67

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 67)

A rocket weighing \(4700\) kg is traveling at a constant \(190\) meters per second. Then when \(t=75200\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(9400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=79900\).

#### Answer.

An IVP modeling this scenario is given by:

\[4700 \, {v'} = 90 \, u\left({t} - 75200\right) - 90 \, u\left({t} - 84600\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{9}{470} \, {\left({t} - 75200\right)} u\left({t} - 75200\right) - \frac{9}{470} \, {\left({t} - 84600\right)} u\left({t} - 84600\right) + 190\]

It follows that when \(t=79900\), the velocity of the rocket is \(280\) meters per second.

#### Example 68

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 68)

A rocket weighing \(4200\) kg is traveling at a constant \(40\) meters per second. Then when \(t=16800\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(12600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=37800\).

#### Answer.

An IVP modeling this scenario is given by:

\[4200 \, {v'} = 90 \, u\left({t} - 16800\right) - 90 \, u\left({t} - 29400\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{3}{140} \, {\left({t} - 16800\right)} u\left({t} - 16800\right) - \frac{3}{140} \, {\left({t} - 29400\right)} u\left({t} - 29400\right) + 40\]

It follows that when \(t=37800\), the velocity of the rocket is \(310\) meters per second.

#### Example 69

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 69)

A rocket weighing \(2800\) kg is traveling at a constant \(90\) meters per second. Then when \(t=28000\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(5600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=30800\).

#### Answer.

An IVP modeling this scenario is given by:

\[2800 \, {v'} = 50 \, u\left({t} - 28000\right) - 50 \, u\left({t} - 33600\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{1}{56} \, {\left({t} - 28000\right)} u\left({t} - 28000\right) - \frac{1}{56} \, {\left({t} - 33600\right)} u\left({t} - 33600\right) + 90\]

It follows that when \(t=30800\), the velocity of the rocket is \(140\) meters per second.

#### Example 70

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 70)

A rocket weighing \(1900\) kg is traveling at a constant \(190\) meters per second. Then when \(t=1900\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(7600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=13300\).

#### Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 10 \, u\left({t} - 1900\right) - 10 \, u\left({t} - 9500\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{1}{190} \, {\left({t} - 1900\right)} u\left({t} - 1900\right) - \frac{1}{190} \, {\left({t} - 9500\right)} u\left({t} - 9500\right) + 190\]

It follows that when \(t=13300\), the velocity of the rocket is \(230\) meters per second.

#### Example 71

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 71)

A rocket weighing \(2100\) kg is traveling at a constant \(170\) meters per second. Then when \(t=10500\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(4200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=12600\).

#### Answer.

An IVP modeling this scenario is given by:

\[2100 \, {v'} = 100 \, u\left({t} - 10500\right) - 100 \, u\left({t} - 14700\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{1}{21} \, {\left({t} - 10500\right)} u\left({t} - 10500\right) - \frac{1}{21} \, {\left({t} - 14700\right)} u\left({t} - 14700\right) + 170\]

It follows that when \(t=12600\), the velocity of the rocket is \(270\) meters per second.

#### Example 72

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 72)

A rocket weighing \(4500\) kg is traveling at a constant \(160\) meters per second. Then when \(t=45000\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(9000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=54000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4500 \, {v'} = 80 \, u\left({t} - 45000\right) - 80 \, u\left({t} - 54000\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{4}{225} \, {\left({t} - 45000\right)} u\left({t} - 45000\right) - \frac{4}{225} \, {\left({t} - 54000\right)} u\left({t} - 54000\right) + 160\]

It follows that when \(t=54000\), the velocity of the rocket is \(320\) meters per second.

#### Example 73

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 73)

A rocket weighing \(2600\) kg is traveling at a constant \(140\) meters per second. Then when \(t=18200\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(7800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=31200\).

#### Answer.

An IVP modeling this scenario is given by:

\[2600 \, {v'} = 50 \, u\left({t} - 18200\right) - 50 \, u\left({t} - 26000\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{1}{52} \, {\left({t} - 18200\right)} u\left({t} - 18200\right) - \frac{1}{52} \, {\left({t} - 26000\right)} u\left({t} - 26000\right) + 140\]

It follows that when \(t=31200\), the velocity of the rocket is \(290\) meters per second.

#### Example 74

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 74)

A rocket weighing \(3300\) kg is traveling at a constant \(50\) meters per second. Then when \(t=56100\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(6600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=62700\).

#### Answer.

An IVP modeling this scenario is given by:

\[3300 \, {v'} = 60 \, u\left({t} - 56100\right) - 60 \, u\left({t} - 62700\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{55} \, {\left({t} - 56100\right)} u\left({t} - 56100\right) - \frac{1}{55} \, {\left({t} - 62700\right)} u\left({t} - 62700\right) + 50\]

It follows that when \(t=62700\), the velocity of the rocket is \(170\) meters per second.

#### Example 75

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 75)

A rocket weighing \(2000\) kg is traveling at a constant \(70\) meters per second. Then when \(t=38000\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(4000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=44000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2000 \, {v'} = 90 \, u\left({t} - 38000\right) - 90 \, u\left({t} - 42000\right)\hspace{2em}v(0)= 70\]

This IVP solves to:

\[{v} = \frac{9}{200} \, {\left({t} - 38000\right)} u\left({t} - 38000\right) - \frac{9}{200} \, {\left({t} - 42000\right)} u\left({t} - 42000\right) + 70\]

It follows that when \(t=44000\), the velocity of the rocket is \(250\) meters per second.

#### Example 76

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 76)

A rocket weighing \(2900\) kg is traveling at a constant \(190\) meters per second. Then when \(t=26100\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(8700\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=43500\).

#### Answer.

An IVP modeling this scenario is given by:

\[2900 \, {v'} = 80 \, u\left({t} - 26100\right) - 80 \, u\left({t} - 34800\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{4}{145} \, {\left({t} - 26100\right)} u\left({t} - 26100\right) - \frac{4}{145} \, {\left({t} - 34800\right)} u\left({t} - 34800\right) + 190\]

It follows that when \(t=43500\), the velocity of the rocket is \(430\) meters per second.

#### Example 77

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 77)

A rocket weighing \(4100\) kg is traveling at a constant \(190\) meters per second. Then when \(t=53300\), its thrusters are turned on, providing \(100\) Newtons of force until they are switched off \(16400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=57400\).

#### Answer.

An IVP modeling this scenario is given by:

\[4100 \, {v'} = 100 \, u\left({t} - 53300\right) - 100 \, u\left({t} - 69700\right)\hspace{2em}v(0)= 190\]

This IVP solves to:

\[{v} = \frac{1}{41} \, {\left({t} - 53300\right)} u\left({t} - 53300\right) - \frac{1}{41} \, {\left({t} - 69700\right)} u\left({t} - 69700\right) + 190\]

It follows that when \(t=57400\), the velocity of the rocket is \(290\) meters per second.

#### Example 78

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 78)

A rocket weighing \(4600\) kg is traveling at a constant \(50\) meters per second. Then when \(t=18400\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(9200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=23000\).

#### Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 70 \, u\left({t} - 18400\right) - 70 \, u\left({t} - 27600\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{7}{460} \, {\left({t} - 18400\right)} u\left({t} - 18400\right) - \frac{7}{460} \, {\left({t} - 27600\right)} u\left({t} - 27600\right) + 50\]

It follows that when \(t=23000\), the velocity of the rocket is \(120\) meters per second.

#### Example 79

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 79)

A rocket weighing \(1600\) kg is traveling at a constant \(60\) meters per second. Then when \(t=19200\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(3200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=20800\).

#### Answer.

An IVP modeling this scenario is given by:

\[1600 \, {v'} = 40 \, u\left({t} - 19200\right) - 40 \, u\left({t} - 22400\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{1}{40} \, {\left({t} - 19200\right)} u\left({t} - 19200\right) - \frac{1}{40} \, {\left({t} - 22400\right)} u\left({t} - 22400\right) + 60\]

It follows that when \(t=20800\), the velocity of the rocket is \(100\) meters per second.

#### Example 80

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 80)

A rocket weighing \(4300\) kg is traveling at a constant \(160\) meters per second. Then when \(t=38700\), its thrusters are turned on, providing \(10\) Newtons of force until they are switched off \(12900\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=64500\).

#### Answer.

An IVP modeling this scenario is given by:

\[4300 \, {v'} = 10 \, u\left({t} - 38700\right) - 10 \, u\left({t} - 51600\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{1}{430} \, {\left({t} - 38700\right)} u\left({t} - 38700\right) - \frac{1}{430} \, {\left({t} - 51600\right)} u\left({t} - 51600\right) + 160\]

It follows that when \(t=64500\), the velocity of the rocket is \(190\) meters per second.

#### Example 81

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 81)

A rocket weighing \(900\) kg is traveling at a constant \(180\) meters per second. Then when \(t=900\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(1800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=1800\).

#### Answer.

An IVP modeling this scenario is given by:

\[900 \, {v'} = 80 \, u\left({t} - 900\right) - 80 \, u\left({t} - 2700\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{4}{45} \, {\left({t} - 900\right)} u\left({t} - 900\right) - \frac{4}{45} \, {\left({t} - 2700\right)} u\left({t} - 2700\right) + 180\]

It follows that when \(t=1800\), the velocity of the rocket is \(260\) meters per second.

#### Example 82

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 82)

A rocket weighing \(1000\) kg is traveling at a constant \(170\) meters per second. Then when \(t=13000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(3000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=18000\).

#### Answer.

An IVP modeling this scenario is given by:

\[1000 \, {v'} = 30 \, u\left({t} - 13000\right) - 30 \, u\left({t} - 16000\right)\hspace{2em}v(0)= 170\]

This IVP solves to:

\[{v} = \frac{3}{100} \, {\left({t} - 13000\right)} u\left({t} - 13000\right) - \frac{3}{100} \, {\left({t} - 16000\right)} u\left({t} - 16000\right) + 170\]

It follows that when \(t=18000\), the velocity of the rocket is \(260\) meters per second.

#### Example 83

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 83)

A rocket weighing \(200\) kg is traveling at a constant \(40\) meters per second. Then when \(t=800\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=1000\).

#### Answer.

An IVP modeling this scenario is given by:

\[200 \, {v'} = 50 \, u\left({t} - 800\right) - 50 \, u\left({t} - 1400\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{1}{4} \, {\left({t} - 800\right)} u\left({t} - 800\right) - \frac{1}{4} \, {\left({t} - 1400\right)} u\left({t} - 1400\right) + 40\]

It follows that when \(t=1000\), the velocity of the rocket is \(90\) meters per second.

#### Example 84

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 84)

A rocket weighing \(2400\) kg is traveling at a constant \(140\) meters per second. Then when \(t=7200\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(4800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=14400\).

#### Answer.

An IVP modeling this scenario is given by:

\[2400 \, {v'} = 80 \, u\left({t} - 7200\right) - 80 \, u\left({t} - 12000\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{1}{30} \, {\left({t} - 7200\right)} u\left({t} - 7200\right) - \frac{1}{30} \, {\left({t} - 12000\right)} u\left({t} - 12000\right) + 140\]

It follows that when \(t=14400\), the velocity of the rocket is \(300\) meters per second.

#### Example 85

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 85)

A rocket weighing \(2500\) kg is traveling at a constant \(80\) meters per second. Then when \(t=30000\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(5000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=40000\).

#### Answer.

An IVP modeling this scenario is given by:

\[2500 \, {v'} = 70 \, u\left({t} - 30000\right) - 70 \, u\left({t} - 35000\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = \frac{7}{250} \, {\left({t} - 30000\right)} u\left({t} - 30000\right) - \frac{7}{250} \, {\left({t} - 35000\right)} u\left({t} - 35000\right) + 80\]

It follows that when \(t=40000\), the velocity of the rocket is \(220\) meters per second.

#### Example 86

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 86)

A rocket weighing \(400\) kg is traveling at a constant \(60\) meters per second. Then when \(t=6000\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(1200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=8000\).

#### Answer.

An IVP modeling this scenario is given by:

\[400 \, {v'} = 30 \, u\left({t} - 6000\right) - 30 \, u\left({t} - 7200\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{3}{40} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) - \frac{3}{40} \, {\left({t} - 7200\right)} u\left({t} - 7200\right) + 60\]

It follows that when \(t=8000\), the velocity of the rocket is \(150\) meters per second.

#### Example 87

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 87)

A rocket weighing \(5000\) kg is traveling at a constant \(100\) meters per second. Then when \(t=100000\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(20000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=110000\).

#### Answer.

An IVP modeling this scenario is given by:

\[5000 \, {v'} = 80 \, u\left({t} - 100000\right) - 80 \, u\left({t} - 120000\right)\hspace{2em}v(0)= 100\]

This IVP solves to:

\[{v} = \frac{2}{125} \, {\left({t} - 100000\right)} u\left({t} - 100000\right) - \frac{2}{125} \, {\left({t} - 120000\right)} u\left({t} - 120000\right) + 100\]

It follows that when \(t=110000\), the velocity of the rocket is \(260\) meters per second.

#### Example 88

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 88)

A rocket weighing \(1300\) kg is traveling at a constant \(160\) meters per second. Then when \(t=19500\), its thrusters are turned on, providing \(30\) Newtons of force until they are switched off \(5200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=28600\).

#### Answer.

An IVP modeling this scenario is given by:

\[1300 \, {v'} = 30 \, u\left({t} - 19500\right) - 30 \, u\left({t} - 24700\right)\hspace{2em}v(0)= 160\]

This IVP solves to:

\[{v} = \frac{3}{130} \, {\left({t} - 19500\right)} u\left({t} - 19500\right) - \frac{3}{130} \, {\left({t} - 24700\right)} u\left({t} - 24700\right) + 160\]

It follows that when \(t=28600\), the velocity of the rocket is \(280\) meters per second.

#### Example 89

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 89)

A rocket weighing \(1100\) kg is traveling at a constant \(180\) meters per second. Then when \(t=2200\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(3300\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=7700\).

#### Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = 70 \, u\left({t} - 2200\right) - 70 \, u\left({t} - 5500\right)\hspace{2em}v(0)= 180\]

This IVP solves to:

\[{v} = \frac{7}{110} \, {\left({t} - 2200\right)} u\left({t} - 2200\right) - \frac{7}{110} \, {\left({t} - 5500\right)} u\left({t} - 5500\right) + 180\]

It follows that when \(t=7700\), the velocity of the rocket is \(390\) meters per second.

#### Example 90

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 90)

A rocket weighing \(4600\) kg is traveling at a constant \(90\) meters per second. Then when \(t=27600\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(18400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=41400\).

#### Answer.

An IVP modeling this scenario is given by:

\[4600 \, {v'} = 20 \, u\left({t} - 27600\right) - 20 \, u\left({t} - 46000\right)\hspace{2em}v(0)= 90\]

This IVP solves to:

\[{v} = \frac{1}{230} \, {\left({t} - 27600\right)} u\left({t} - 27600\right) - \frac{1}{230} \, {\left({t} - 46000\right)} u\left({t} - 46000\right) + 90\]

It follows that when \(t=41400\), the velocity of the rocket is \(150\) meters per second.

#### Example 91

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 91)

A rocket weighing \(200\) kg is traveling at a constant \(140\) meters per second. Then when \(t=3400\), its thrusters are turned on, providing \(70\) Newtons of force until they are switched off \(600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=4400\).

#### Answer.

An IVP modeling this scenario is given by:

\[200 \, {v'} = 70 \, u\left({t} - 3400\right) - 70 \, u\left({t} - 4000\right)\hspace{2em}v(0)= 140\]

This IVP solves to:

\[{v} = \frac{7}{20} \, {\left({t} - 3400\right)} u\left({t} - 3400\right) - \frac{7}{20} \, {\left({t} - 4000\right)} u\left({t} - 4000\right) + 140\]

It follows that when \(t=4400\), the velocity of the rocket is \(350\) meters per second.

#### Example 92

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 92)

A rocket weighing \(3700\) kg is traveling at a constant \(40\) meters per second. Then when \(t=66600\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(7400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=77700\).

#### Answer.

An IVP modeling this scenario is given by:

\[3700 \, {v'} = 20 \, u\left({t} - 66600\right) - 20 \, u\left({t} - 74000\right)\hspace{2em}v(0)= 40\]

This IVP solves to:

\[{v} = \frac{1}{185} \, {\left({t} - 66600\right)} u\left({t} - 66600\right) - \frac{1}{185} \, {\left({t} - 74000\right)} u\left({t} - 74000\right) + 40\]

It follows that when \(t=77700\), the velocity of the rocket is \(80\) meters per second.

#### Example 93

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 93)

A rocket weighing \(2600\) kg is traveling at a constant \(20\) meters per second. Then when \(t=7800\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(7800\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=23400\).

#### Answer.

An IVP modeling this scenario is given by:

\[2600 \, {v'} = 80 \, u\left({t} - 7800\right) - 80 \, u\left({t} - 15600\right)\hspace{2em}v(0)= 20\]

This IVP solves to:

\[{v} = \frac{2}{65} \, {\left({t} - 7800\right)} u\left({t} - 7800\right) - \frac{2}{65} \, {\left({t} - 15600\right)} u\left({t} - 15600\right) + 20\]

It follows that when \(t=23400\), the velocity of the rocket is \(260\) meters per second.

#### Example 94

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 94)

A rocket weighing \(1100\) kg is traveling at a constant \(150\) meters per second. Then when \(t=4400\), its thrusters are turned on, providing \(60\) Newtons of force until they are switched off \(4400\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=9900\).

#### Answer.

An IVP modeling this scenario is given by:

\[1100 \, {v'} = 60 \, u\left({t} - 4400\right) - 60 \, u\left({t} - 8800\right)\hspace{2em}v(0)= 150\]

This IVP solves to:

\[{v} = \frac{3}{55} \, {\left({t} - 4400\right)} u\left({t} - 4400\right) - \frac{3}{55} \, {\left({t} - 8800\right)} u\left({t} - 8800\right) + 150\]

It follows that when \(t=9900\), the velocity of the rocket is \(390\) meters per second.

#### Example 95

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 95)

A rocket weighing \(4300\) kg is traveling at a constant \(60\) meters per second. Then when \(t=51600\), its thrusters are turned on, providing \(80\) Newtons of force until they are switched off \(8600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=60200\).

#### Answer.

An IVP modeling this scenario is given by:

\[4300 \, {v'} = 80 \, u\left({t} - 51600\right) - 80 \, u\left({t} - 60200\right)\hspace{2em}v(0)= 60\]

This IVP solves to:

\[{v} = \frac{4}{215} \, {\left({t} - 51600\right)} u\left({t} - 51600\right) - \frac{4}{215} \, {\left({t} - 60200\right)} u\left({t} - 60200\right) + 60\]

It follows that when \(t=60200\), the velocity of the rocket is \(220\) meters per second.

#### Example 96

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 96)

A rocket weighing \(3500\) kg is traveling at a constant \(150\) meters per second. Then when \(t=3500\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(14000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=28000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3500 \, {v'} = 50 \, u\left({t} - 3500\right) - 50 \, u\left({t} - 17500\right)\hspace{2em}v(0)= 150\]

This IVP solves to:

\[{v} = \frac{1}{70} \, {\left({t} - 3500\right)} u\left({t} - 3500\right) - \frac{1}{70} \, {\left({t} - 17500\right)} u\left({t} - 17500\right) + 150\]

It follows that when \(t=28000\), the velocity of the rocket is \(350\) meters per second.

#### Example 97

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 97)

A rocket weighing \(2400\) kg is traveling at a constant \(30\) meters per second. Then when \(t=28800\), its thrusters are turned on, providing \(90\) Newtons of force until they are switched off \(9600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=43200\).

#### Answer.

An IVP modeling this scenario is given by:

\[2400 \, {v'} = 90 \, u\left({t} - 28800\right) - 90 \, u\left({t} - 38400\right)\hspace{2em}v(0)= 30\]

This IVP solves to:

\[{v} = \frac{3}{80} \, {\left({t} - 28800\right)} u\left({t} - 28800\right) - \frac{3}{80} \, {\left({t} - 38400\right)} u\left({t} - 38400\right) + 30\]

It follows that when \(t=43200\), the velocity of the rocket is \(390\) meters per second.

#### Example 98

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 98)

A rocket weighing \(400\) kg is traveling at a constant \(50\) meters per second. Then when \(t=4800\), its thrusters are turned on, providing \(40\) Newtons of force until they are switched off \(1200\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=5200\).

#### Answer.

An IVP modeling this scenario is given by:

\[400 \, {v'} = 40 \, u\left({t} - 4800\right) - 40 \, u\left({t} - 6000\right)\hspace{2em}v(0)= 50\]

This IVP solves to:

\[{v} = \frac{1}{10} \, {\left({t} - 4800\right)} u\left({t} - 4800\right) - \frac{1}{10} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) + 50\]

It follows that when \(t=5200\), the velocity of the rocket is \(90\) meters per second.

#### Example 99

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 99)

A rocket weighing \(1900\) kg is traveling at a constant \(20\) meters per second. Then when \(t=36100\), its thrusters are turned on, providing \(50\) Newtons of force until they are switched off \(7600\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=43700\).

#### Answer.

An IVP modeling this scenario is given by:

\[1900 \, {v'} = 50 \, u\left({t} - 36100\right) - 50 \, u\left({t} - 43700\right)\hspace{2em}v(0)= 20\]

This IVP solves to:

\[{v} = \frac{1}{38} \, {\left({t} - 36100\right)} u\left({t} - 36100\right) - \frac{1}{38} \, {\left({t} - 43700\right)} u\left({t} - 43700\right) + 20\]

It follows that when \(t=43700\), the velocity of the rocket is \(220\) meters per second.

#### Example 100

## D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration. (ver. 100)

A rocket weighing \(3000\) kg is traveling at a constant \(80\) meters per second. Then when \(t=0\), its thrusters are turned on, providing \(20\) Newtons of force until they are switched off \(6000\) seconds later.

Give an IVP that models this scenario, then solve it. Use your solution to find the velocity of the rocket when \(t=6000\).

#### Answer.

An IVP modeling this scenario is given by:

\[3000 \, {v'} = -20 \, u\left({t} - 6000\right) + 20 \, u\left({t}\right)\hspace{2em}v(0)= 80\]

This IVP solves to:

\[{v} = -\frac{1}{150} \, {\left({t} - 6000\right)} u\left({t} - 6000\right) + \frac{1}{150} \, {t} u\left({t}\right) + 80\]

It follows that when \(t=6000\), the velocity of the rocket is \(120\) meters per second.