# Section 6.5 – Impulse Functions

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Find the solution of the initial value problem ${y}^{″}-y=4\delta \left(t-2\right)+{t}^{2},\phantom{\rule{2em}{0ex}}y\left(0\right)=0,\phantom{\rule{1em}{0ex}}{y}^{\prime }\left(0\right)=2$

$$x(t)=-2-t^2+2\cosh(t)+2\sinh(t)+4u_2(t)\sinh(t-2)$$

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2. A 2-kg mass is attached to a spring with the Hooke's constant of 3 N/m, and it is subject to moving in a medium with a damping constant of 5 N-s/m. The mass is initially displaced +0.5 m from
its equilibrium position and is released. Then after 2 seconds, a hammer hits the mass in such a way that its velocity suddenly changes by 5m/s in the positive direction (i.e., $\mathrm{\Delta }v=+5$ m/s).
1. Find the impulse of the hammer and its units.
2. Write the hammer force using an appropriate Dirac delta function. Note that your force must result in the same impulse value you found in part (a).
3. Set up an IVP and find the position $x\left(t\right)$ of the mass for $t\ge 0.$

1. $$I=10\ N\cdot s$$
2. $$F(t)=10\delta(t-2)$$
3. $$x(t)=-e^{-\frac{3}{2}t}+1.5e^{-t}+u_2(t)\left(10e^{-(t-2)}-10e^{-\frac{3}{2}(t-2)}\right)$$

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