Often bumps like the one depicted below 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 speed V is given by y(t) = 0 for t lessthanorequalto -L/2V my" + ky = {cos (pi Vt/L) for -L/2V lessthanorequalto t lessthanorequalto L/2V 0 for t greaterthanorequalto L/2V (The absence of a damping term indicates that the car's shock absorbers are broken.) Note that the equations are dependent on time only; as the speed is given as V, we can write space x in terms of time t: x = Vt. (a) Solve this initial value problem; take m = k = 1 and L = pi for convenience. Thus show that the formula for oscillatory motion after the car has traversed the speed bump is y(t) = A sin(t), where A depends on the speed V (b) Use Matlab/Octave (or similar) to plot the amplitude |A| of the solution y (t) in part (a) versus the car's speed V. From the graph, estimate the speed that produces the most violent shaking of the vehicle