Speed Bumps. Often speed bumps are built into roads to discourage speeding. The

Question

Speed Bumps. Often speed bumps are built into roads to discourage speeding. The figure suggests that a crude model of the vertical motion y(t) of a car encountering the speed bump with the speed V is given by

Y(t)=0 for t is less then or equal to -L/(2V),

my"+ky = F*cos(pi*Vt/L) when t < L/(2V) or

my"+ky = 0 when t greater than or equal to L/(2V)

the absence of a damping term indicates that the car's shock absorbers are not functioning.

A) taking m=k=1 and L=pi and F=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(t)=Asin(t), where the constant A depends on the speed V.

Explanation / Answer

For the homogenous part, we use characteristics equation where mr^2 + k = 0, ie, r = sqrt{k/m}*i, where i = sqrt{-1}. Let's call w = sqrt{k/m}. Then the homogeneous solution to the problem is y = Asin(wt) + B cos(wt). Plug in y(-L/2V) = 0 = y(L/2V) to find the constants A and B.

For the particular part, we have to guess the solution. For a cosine function being the force, we guess yp = Ccos(pi*Vt/L) + Dsin(pi*Vt/L). We want to plug this into the differential equation, so we differentiate to get yp`` = -C*(pi*V/L)^2 cos(pi*Vt/L) - D*(pi*V/L)^2 sin(pi*Vt/L). Plug yp and yp`` into the differential equation to get m*yp`` + k*yp = -Cm*(pi*V/L)^2 cos(pi*Vt/L) - Dk*(pi*V/L)^2 sin(pi*Vt/L) + Cm*cos(pi*Vt/L) + Dk*sin(pi*Vt/L) = cos(pi*Vt/L). By matching the coefficients of the sine and the cosine, we find that D = 0, and that -Cm*(pi*V/L)^2 + Cm = 1. We can solve for C.

The final answer to the question is y + yp.

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