Chapter 4.5, Problem 45E

Speed Bumps. Often bumps like the one depicted in Figure 4.11 are built into roads to discourage speeding. The figure suggests that a crude model of the vertical motion $y\left(t\right)$ of a car encountering the speed bump with the speed $V$ is given by

$\begin{array}{rll}y\left(t\right)& =& 0,\text{ for }t\le -L/2V\\ m{y}^{″}+ky& =& \left\{\begin{array}{l}{F}_0\cos \left(\pi Vt/L\right)\text{ for }\left|t\right|<L/\left(2V\right)\\ 0\text{ for }t\ge L/\left(2V\right)\end{array}\right\}\end{array}$

(The absence of a damping term indicates that the car’s shock absorbers are not functioning.)

a. Taking $m=k=1$, $L=\pi$, and $F_0=1$ in appropriate units, solve this initial value problem. Thereby show that the formula for the oscillatory motion after the car has traversed the speed bump is $y\left(t\right)=A\sin t$ where the constant $A$ depends on the speed $V$.

b. Plot the amplitude $\left|A\right|$ of the solution $y\left(t\right)$ found in part (a) versus the car’s speed $V$. From the graph, estimate the speed that produces the most violent shaking of the vehicle.