Sample Exam 2 w Solutions (2011) – Differential Equations

y(s)LasFigure 1: Bumpy road.Problem 1a. (12 pts) A car is moving at constant speed v on a road which becomes bumpy.The equation of the surface is y(s) = asin( 2πs)= L 0.1sin(2πs ) where a = 0.1 m is the amplitude6of the bumps and L = 6 m is their wavelength (figure 1).Assume the car behaves like a mass of 1,000 kg on a spring with spring constant 100,000 N/m,without shock absorber. We want to know how the bumpy road will affect the car. Let x(t)be the upward displacement of the car, relative to its static equilibrium position – i.e. theequilibrium position once the spring has been compressed by the weight of the car.Find x(t), assuming the car was always at its vertical equilibrium position before reaching thefirst bump (at time t = 0). Assume a generic value of the speed v. Hint: The spring is stretchedby x−y and the spring force is the only force to take into account.The weight of the car is taken into account by changing the reference to the static equilibriumposition. The spring force is (effectively) k(x−y) = 100,000(x−y) and so Newton’s law appliedto the vertical position of the car yields:mx ′′ = −k(x−y)1,000x ′′ = −100,000(x−y).The horizontal position of the car is s(t) = vt, at which the elevation of the road is y(t) =0.1sin( πvt ) so the equation becomes:3(1,000x ′′ = −100,000 x−0.1sin πvt )3x ′′ = −100x+10sin πvt3x ′′ +100x = 10sin πvt3 .The homogeneous equation x ′′ +100x = 0 has a basis of solutions cos10t, sin10t. Let us finda solution of the non-homogeneous equation of the form:2