## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

## 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.

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:

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

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.

## 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.

## 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.

## 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.

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:

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

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.

## 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.

## 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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.

## 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.

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:

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

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.