Suppose that a car oscillates vertically as if it were a mass of m = 800kg on a single spring with spring constant k = 7*10^4 N/m which is attached to a single damper with c = 3000 N*s/m. Suppose that this car is driven along a sinusoidal road surface with an amplitude of 5 cm and a wavelength of L = 10m.
1. Write the non-homogeneous ODE that governs the vertical motion of the car.
(Make sure to keep track of units).
2. Find the steady-state amplitude of vibration of the car as a function of the horizontal driving speed v (m/s).
3. At what driving speed v_m will the greatest amplitude of vertical vibrations occur?
4. If the driver would prefer that the car oscillates with a lower amplitude than that of the underlying road surface (i.e. A < 5cm), in what range of speeds must they drive?