Meriam, Kraige & Bolton — Kinematics of Particles: Problem 2_32

2/32 The falling object has a speed \(v_0\) when it strikes and subsequently deforms the foam arresting material until it comes to rest. The resistance of the foam material to deformation is a function of penetration depth \(y\) and object speed \(v\) so that the acceleration of the object is \(a = g - k_1v - k_2y\), where \(v\) is the particle speed in inches per second, \(y\) is the penetration depth in inches, and \(k_1\) and \(k_2\) are positive constants. Plot the penetration depth \(y\) and velocity \(v\) of the object as functions of time over the first five seconds for \(k_1 = 12\text{ sec}^{-1}\), \(k_2 = 24\text{ sec}^{-2}\), and \(v_0 = 25\text{ in./sec}\). Determine the time when the penetration depth reaches 95% of its final value.