## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse.

#### Example 1

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 1)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=18\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -5 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=18\), the position of the mass is \(-0.5275\) meters.

#### Example 2

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 2)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=10\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -2 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{15} \, \sin\left(3 \, {t} - 9\right) u\left({t} - 3\right) + 4 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(0.5054\) meters.

#### Example 3

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 3)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=16\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -7 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(4 \, {t} - 32\right) u\left({t} - 8\right) + 5 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(1.638\) meters.

#### Example 4

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 4)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=10\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -3 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{8} \, \sin\left(2 \, {t} - 18\right) u\left({t} - 9\right) + 6 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(2.108\) meters.

#### Example 5

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 5)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=9\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -3 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 4 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(-1.375\) meters.

#### Example 6

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 6)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=12\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -4 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{5} \, \sin\left(4 \, {t} - 28\right) u\left({t} - 7\right) + 3 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-2.103\) meters.

#### Example 7

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 7)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(10\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=14\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -5 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=10,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{4} \, \sin\left(4 \, {t} - 32\right) u\left({t} - 8\right) + 10 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(8.759\) meters.

#### Example 8

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 8)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=20\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -6 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{10} \, \sin\left(4 \, {t} - 40\right) u\left({t} - 10\right) + 2 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=20\), the position of the mass is \(-0.4443\) meters.

#### Example 9

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 9)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=8\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -5 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{8} \, \sin\left(2 \, {t} - 6\right) u\left({t} - 3\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(-3.491\) meters.

#### Example 10

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 10)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=12\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -5 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-0.03483\) meters.

#### Example 11

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 11)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=5\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -3 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 9\right) u\left({t} - 3\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-1.426\) meters.

#### Example 12

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 12)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=14\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -5 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{9} \, \sin\left(3 \, {t} - 15\right) u\left({t} - 5\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(-4.131\) meters.

#### Example 13

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 13)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=6\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -2 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{4} \, \sin\left(4 \, {t} - 8\right) u\left({t} - 2\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=6\), the position of the mass is \(3.465\) meters.

#### Example 14

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 14)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=11\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -8 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -2 \, \sin\left(2 \, {t} - 10\right) u\left({t} - 5\right) + 3 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(-1.927\) meters.

#### Example 15

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 15)

A \(4\) kg mass is attached to a spring with constant \(64\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=9\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 64 \, {x} = -3 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{16} \, \sin\left(4 \, {t} - 28\right) u\left({t} - 7\right) + 7 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(-1.081\) meters.

#### Example 16

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 16)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=10\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -2 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{9} \, \sin\left(3 \, {t} - 3\right) u\left({t} - 1\right) + 8 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(1.021\) meters.

#### Example 17

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 17)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=13\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -5 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{9} \, \sin\left(3 \, {t} - 15\right) u\left({t} - 5\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(2.903\) meters.

#### Example 18

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 18)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=15\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -6 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{5} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + 3 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(1.316\) meters.

#### Example 19

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 19)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=6\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -8 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(4 \, {t} - 12\right) u\left({t} - 3\right) + 6 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=6\), the position of the mass is \(2.903\) meters.

#### Example 20

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 20)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=11\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -6 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(3 \, {t} - 27\right) u\left({t} - 9\right) + 8 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(0.08006\) meters.

#### Example 21

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 21)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=14\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -3 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{4} \, \sin\left(2 \, {t} - 20\right) u\left({t} - 10\right) + 7 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(-7.480\) meters.

#### Example 22

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 22)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=20\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -3 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=20\), the position of the mass is \(-0.6231\) meters.

#### Example 23

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 23)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=11\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -3 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{4} \, \sin\left(2 \, {t} - 18\right) u\left({t} - 9\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(-3.432\) meters.

#### Example 24

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 24)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=13\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -4 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(4 \, {t} - 36\right) u\left({t} - 9\right) + 6 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(-0.8820\) meters.

#### Example 25

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 25)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -8 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(4 \, {t} - 16\right) u\left({t} - 4\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(6.329\) meters.

#### Example 26

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 26)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=16\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -2 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{10} \, \sin\left(4 \, {t} - 24\right) u\left({t} - 6\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(3.060\) meters.

#### Example 27

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 27)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=8\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -8 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(4 \, {t} - 4\right) u\left({t} - 1\right) + 4 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(3.066\) meters.

#### Example 28

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 28)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -7 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(4 \, {t} - 12\right) u\left({t} - 3\right) + 4 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(3.678\) meters.

#### Example 29

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 29)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=12\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -4 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{4}{15} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(0.07230\) meters.

#### Example 30

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 30)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=8\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -2 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{6} \, \sin\left(4 \, {t} - 28\right) u\left({t} - 7\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(6.800\) meters.

#### Example 31

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 31)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -6 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(3 \, {t} - 27\right) u\left({t} - 9\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(-1.050\) meters.

#### Example 32

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 32)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=13\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -7 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(3 \, {t} - 9\right) u\left({t} - 3\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(2.976\) meters.

#### Example 33

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 33)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=9\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -7 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{6} \, \sin\left(3 \, {t} - 3\right) u\left({t} - 1\right) + 8 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(-1.281\) meters.

#### Example 34

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 34)

A \(4\) kg mass is attached to a spring with constant \(64\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=18\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 64 \, {x} = -4 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{4} \, \sin\left(4 \, {t} - 36\right) u\left({t} - 9\right) + 4 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=18\), the position of the mass is \(-3.621\) meters.

#### Example 35

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 35)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(10\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=18\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -5 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=10,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{12} \, \sin\left(4 \, {t} - 32\right) u\left({t} - 8\right) + 10 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=18\), the position of the mass is \(-9.983\) meters.

#### Example 36

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 36)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=5\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -5 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 12\right) u\left({t} - 4\right) + 3 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-2.326\) meters.

#### Example 37

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 37)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -8 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{5} \, \sin\left(4 \, {t} - 16\right) u\left({t} - 4\right) + 3 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(-1.639\) meters.

#### Example 38

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 38)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -2 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(2 \, {t} - 2\right) u\left({t} - 1\right) + 8 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(-8.456\) meters.

#### Example 39

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 39)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=4\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -6 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{5} \, \sin\left(2 \, {t} - 2\right) u\left({t} - 1\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=4\), the position of the mass is \(-0.4144\) meters.

#### Example 40

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 40)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -2 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{9} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(0.3474\) meters.

#### Example 41

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 41)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=15\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -8 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(4 \, {t} - 20\right) u\left({t} - 5\right) + 3 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(-3.354\) meters.

#### Example 42

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 42)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -7 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(4 \, {t} - 16\right) u\left({t} - 4\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(6.842\) meters.

#### Example 43

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 43)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=15\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -5 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{8} \, \sin\left(4 \, {t} - 24\right) u\left({t} - 6\right) + 7 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(-6.047\) meters.

#### Example 44

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 44)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -2 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(2 \, {t} - 12\right) u\left({t} - 6\right) + \cos\left(2 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(-0.08660\) meters.

#### Example 45

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 45)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=5\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -2 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{15} \, \sin\left(3 \, {t} - 6\right) u\left({t} - 2\right) + 7 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-5.373\) meters.

#### Example 46

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 46)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -4 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(4 \, {t} - 40\right) u\left({t} - 10\right) + 4 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-2.890\) meters.

#### Example 47

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 47)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=4\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -2 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{15} \, \sin\left(3 \, {t} - 3\right) u\left({t} - 1\right) + 3 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=4\), the position of the mass is \(2.477\) meters.

#### Example 48

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 48)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -6 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(3 \, {t} - 24\right) u\left({t} - 8\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(3.891\) meters.

#### Example 49

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 49)

A \(3\) kg mass is attached to a spring with constant \(12\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 12 \, {x} = -3 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(2 \, {t} - 4\right) u\left({t} - 2\right) + 5 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(1.664\) meters.

#### Example 50

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 50)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -8 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(2 \, {t} - 10\right) u\left({t} - 5\right) + 7 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(3.633\) meters.

#### Example 51

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 51)

A \(3\) kg mass is attached to a spring with constant \(12\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 12 \, {x} = -6 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(2 \, {t} - 16\right) u\left({t} - 8\right) + 6 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=18\), the position of the mass is \(-1.681\) meters.

#### Example 52

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 52)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(10\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -5 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=10,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{12} \, \sin\left(4 \, {t} - 24\right) u\left({t} - 6\right) + 10 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-6.024\) meters.

#### Example 53

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 53)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -7 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(3 \, {t} - 21\right) u\left({t} - 7\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(0.7660\) meters.

#### Example 54

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 54)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -5 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{12} \, \sin\left(3 \, {t} - 27\right) u\left({t} - 9\right) + 8 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(0.01021\) meters.

#### Example 55

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 55)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -2 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{10} \, \sin\left(4 \, {t} - 28\right) u\left({t} - 7\right) + 3 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-2.012\) meters.

#### Example 56

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 56)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -7 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{8} \, \sin\left(2 \, {t} - 8\right) u\left({t} - 4\right) + 6 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(2.797\) meters.

#### Example 57

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 57)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -7 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{8} \, \sin\left(2 \, {t} - 14\right) u\left({t} - 7\right) + 7 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(1.332\) meters.

#### Example 58

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 58)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -4 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{3} \, \sin\left(3 \, {t} - 12\right) u\left({t} - 4\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-0.8067\) meters.

#### Example 59

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 59)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -7 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{4} \, \sin\left(2 \, {t} - 18\right) u\left({t} - 9\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(0.8563\) meters.

#### Example 60

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 60)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=16\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -2 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{9} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 4 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(-2.341\) meters.

#### Example 61

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 61)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -4 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(4 \, {t} - 32\right) u\left({t} - 8\right) + 9 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(8.132\) meters.

#### Example 62

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 62)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=6\).

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -3 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(3 \, {t} - 9\right) u\left({t} - 3\right) + 6 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=6\), the position of the mass is \(3.756\) meters.

#### Example 63

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 63)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=7\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -3 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{10} \, \sin\left(2 \, {t} - 8\right) u\left({t} - 4\right) + 6 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=7\), the position of the mass is \(0.9043\) meters.

#### Example 64

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 64)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -2 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{10} \, \sin\left(4 \, {t} - 20\right) u\left({t} - 5\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=13\), the position of the mass is \(-1.359\) meters.

#### Example 65

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 65)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=19\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -5 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(2 \, {t} - 20\right) u\left({t} - 10\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=19\), the position of the mass is \(4.196\) meters.

#### Example 66

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 66)

A \(3\) kg mass is attached to a spring with constant \(12\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=7\).

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 12 \, {x} = -8 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{4}{3} \, \sin\left(2 \, {t} - 4\right) u\left({t} - 2\right) + \cos\left(2 \, {t}\right)\]

It follows that when \(t=7\), the position of the mass is \(0.8621\) meters.

#### Example 67

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 67)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -8 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{5} \, \sin\left(4 \, {t} - 36\right) u\left({t} - 9\right) + 2 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(-1.031\) meters.

#### Example 68

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 68)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -7 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{8} \, \sin\left(4 \, {t} - 12\right) u\left({t} - 3\right) + 6 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(5.517\) meters.

#### Example 69

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 69)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -5 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{9} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(1.004\) meters.

#### Example 70

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 70)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -2 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{15} \, \sin\left(3 \, {t} - 3\right) u\left({t} - 1\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(-2.508\) meters.

#### Example 71

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 71)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -8 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{8}{15} \, \sin\left(3 \, {t} - 9\right) u\left({t} - 3\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-1.370\) meters.

#### Example 72

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 72)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -7 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{20} \, \sin\left(4 \, {t} - 20\right) u\left({t} - 5\right) + 3 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(-2.320\) meters.

#### Example 73

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 73)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(4\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -5 \, \delta\left({t} - 4\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{12} \, \sin\left(3 \, {t} - 12\right) u\left({t} - 4\right) + 6 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-0.3905\) meters.

#### Example 74

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 74)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -5 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{6} \, \sin\left(3 \, {t} - 27\right) u\left({t} - 9\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(-1.342\) meters.

#### Example 75

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 75)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -7 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{9} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + 3 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(-1.336\) meters.

#### Example 76

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 76)

A \(5\) kg mass is attached to a spring with constant \(45\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 45 \, {x} = -7 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{15} \, \sin\left(3 \, {t} - 15\right) u\left({t} - 5\right) + 6 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(3.613\) meters.

#### Example 77

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 77)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -8 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{5} \, \sin\left(4 \, {t} - 28\right) u\left({t} - 7\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(6.977\) meters.

#### Example 78

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 78)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -6 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{4} \, \sin\left(4 \, {t} - 12\right) u\left({t} - 3\right) + \cos\left(4 \, {t}\right)\]

It follows that when \(t=6\), the position of the mass is \(0.8266\) meters.

#### Example 79

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 79)

A \(3\) kg mass is attached to a spring with constant \(12\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 12 \, {x} = -4 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(2 \, {t} - 14\right) u\left({t} - 7\right) + 3 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(-3.479\) meters.

#### Example 80

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 80)

A \(5\) kg mass is attached to a spring with constant \(80\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(5\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 80 \, {x} = -2 \, \delta\left({t} - 5\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{10} \, \sin\left(4 \, {t} - 20\right) u\left({t} - 5\right) + 5 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(-4.837\) meters.

#### Example 81

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 81)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(3\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=3\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -7 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=3,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{10} \, \sin\left(2 \, {t} - 2\right) u\left({t} - 1\right) + 3 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=3\), the position of the mass is \(3.410\) meters.

#### Example 82

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 82)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -3 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{10} \, \sin\left(2 \, {t} - 14\right) u\left({t} - 7\right) + 4 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(3.562\) meters.

#### Example 83

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 83)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -5 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{4} \, \sin\left(2 \, {t} - 6\right) u\left({t} - 3\right) + 5 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-3.249\) meters.

#### Example 84

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 84)

A \(4\) kg mass is attached to a spring with constant \(36\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 36 \, {x} = -7 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(3 \, {t} - 6\right) u\left({t} - 2\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(1.286\) meters.

#### Example 85

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 85)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -6 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{3} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 2 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(0.4131\) meters.

#### Example 86

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 86)

A \(3\) kg mass is attached to a spring with constant \(12\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 12 \, {x} = -3 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{2} \, \sin\left(2 \, {t} - 16\right) u\left({t} - 8\right) + 7 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(5.984\) meters.

#### Example 87

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 87)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -7 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(4 \, {t} - 40\right) u\left({t} - 10\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(-5.698\) meters.

#### Example 88

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 88)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(10\) meters and released from rest. Then after \(8\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=17\).

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -4 \, \delta\left({t} - 8\right)\hspace{2em}x(0)=10,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{5} \, \sin\left(2 \, {t} - 16\right) u\left({t} - 8\right) + 10 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=17\), the position of the mass is \(-8.185\) meters.

#### Example 89

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 89)

A \(5\) kg mass is attached to a spring with constant \(20\) N/m. The mass is pulled outward \(6\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[5 \, {x''} + 20 \, {x} = -6 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=6,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{5} \, \sin\left(2 \, {t} - 4\right) u\left({t} - 2\right) + 6 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=10\), the position of the mass is \(2.621\) meters.

#### Example 90

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 90)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(10\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -3 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=10,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{1}{4} \, \sin\left(4 \, {t} - 12\right) u\left({t} - 3\right) + 10 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=6\), the position of the mass is \(4.376\) meters.

#### Example 91

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 91)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(6\) Newtons of inward impulse.

Give an IVP that models this scenario, then solve it. Use your solution to find the position of the mass when \(t=17\).

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -6 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{4} \, \sin\left(2 \, {t} - 18\right) u\left({t} - 9\right) + 5 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=17\), the position of the mass is \(-4.027\) meters.

#### Example 92

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 92)

A \(2\) kg mass is attached to a spring with constant \(32\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(9\) seconds, the mass is struck by a hammer, imparting \(3\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 32 \, {x} = -3 \, \delta\left({t} - 9\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{3}{8} \, \sin\left(4 \, {t} - 36\right) u\left({t} - 9\right) + 2 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=16\), the position of the mass is \(0.6821\) meters.

#### Example 93

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 93)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(4\) meters and released from rest. Then after \(2\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -7 \, \delta\left({t} - 2\right)\hspace{2em}x(0)=4,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{6} \, \sin\left(3 \, {t} - 6\right) u\left({t} - 2\right) + 4 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=12\), the position of the mass is \(0.6409\) meters.

#### Example 94

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 94)

A \(4\) kg mass is attached to a spring with constant \(16\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(3\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 16 \, {x} = -5 \, \delta\left({t} - 3\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{8} \, \sin\left(2 \, {t} - 6\right) u\left({t} - 3\right) + 9 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(6.278\) meters.

#### Example 95

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 95)

A \(3\) kg mass is attached to a spring with constant \(48\) N/m. The mass is pulled outward \(2\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(7\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 48 \, {x} = -7 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=2,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{7}{12} \, \sin\left(4 \, {t} - 24\right) u\left({t} - 6\right) + 2 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=11\), the position of the mass is \(1.467\) meters.

#### Example 96

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 96)

A \(4\) kg mass is attached to a spring with constant \(64\) N/m. The mass is pulled outward \(8\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[4 \, {x''} + 64 \, {x} = -5 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=8,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{16} \, \sin\left(4 \, {t} - 4\right) u\left({t} - 1\right) + 8 \, \cos\left(4 \, {t}\right)\]

It follows that when \(t=9\), the position of the mass is \(-1.196\) meters.

#### Example 97

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 97)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(7\) meters and released from rest. Then after \(6\) seconds, the mass is struck by a hammer, imparting \(8\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -8 \, \delta\left({t} - 6\right)\hspace{2em}x(0)=7,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{4}{3} \, \sin\left(3 \, {t} - 18\right) u\left({t} - 6\right) + 7 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=14\), the position of the mass is \(-1.592\) meters.

#### Example 98

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 98)

A \(2\) kg mass is attached to a spring with constant \(8\) N/m. The mass is pulled outward \(5\) meters and released from rest. Then after \(7\) seconds, the mass is struck by a hammer, imparting \(4\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 8 \, {x} = -4 \, \delta\left({t} - 7\right)\hspace{2em}x(0)=5,x'(0)=0\]

This IVP solves to:

\[{x} = -\sin\left(2 \, {t} - 14\right) u\left({t} - 7\right) + 5 \, \cos\left(2 \, {t}\right)\]

It follows that when \(t=8\), the position of the mass is \(-5.698\) meters.

#### Example 99

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 99)

A \(2\) kg mass is attached to a spring with constant \(18\) N/m. The mass is pulled outward \(9\) meters and released from rest. Then after \(10\) seconds, the mass is struck by a hammer, imparting \(5\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[2 \, {x''} + 18 \, {x} = -5 \, \delta\left({t} - 10\right)\hspace{2em}x(0)=9,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{5}{6} \, \sin\left(3 \, {t} - 30\right) u\left({t} - 10\right) + 9 \, \cos\left(3 \, {t}\right)\]

It follows that when \(t=15\), the position of the mass is \(4.186\) meters.

#### Example 100

## D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse. (ver. 100)

A \(3\) kg mass is attached to a spring with constant \(27\) N/m. The mass is pulled outward \(1\) meters and released from rest. Then after \(1\) seconds, the mass is struck by a hammer, imparting \(2\) Newtons of inward impulse.

#### Answer.

An IVP modeling this scenario is given by:

\[3 \, {x''} + 27 \, {x} = -2 \, \delta\left({t} - 1\right)\hspace{2em}x(0)=1,x'(0)=0\]

This IVP solves to:

\[{x} = -\frac{2}{9} \, \sin\left(3 \, {t} - 3\right) u\left({t} - 1\right) + \cos\left(3 \, {t}\right)\]

It follows that when \(t=5\), the position of the mass is \(-0.6404\) meters.