Let Y have density function {ove ² [cye 2, y 20 0, otherwise f(y)=

This problem is an example of critically damped harmonic motion. A mass m = 4 kg is attached to both a spring with spring constant k = 144 N/m and a dash-pot with damping constant c = 48 N-s/m. The ball is started in motion with initial position = 3 m and initial velocity = -20 m/s. Determine the position function z(t) in meters. z(t) = Graph the function (t). Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function (). In this case the position function u(1) can be written as u(1) - Cocos (wlan). Determine Co. wo and an 5 3 Co- والها 8 (assume 0 < a < 2π) Finally, graph both function (t) and u(t) in the same window to illustrate the effect of damping. 11 ap

A random variable, H, has the moment generating function, My(t) = e¹2e¹-12. De- termine the second moment of the random variable H, E[H²].

Part b) Find the second moment of Y using E[Y²] = E, y²p(y) (finding the first moment using any other method will result in no credit, so do not use a moment generating function).