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.