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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.

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:

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

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.