A certain model of the motion of a light plastic ball tossed into the air is given by

\[m x^{\prime \prime}+c x^{\prime}+m g=0, \quad x(0)=0, \quad x^{\prime}(0)=v_{0}\]

Here \(m\) is the mass of the ball, \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(c\) is a measure of the damping. Because there is no \(x\) term, we can write this as a first order equation for the velocity \(v(t)=x^{\prime}(t)\) :

\[m v^{\prime}+c v+m g=0\]

a. Find the general solution for the velocity \(v(t)\) of the linear firstorder differential equation above.

b. Use the solution of part a to find the general solution for the position \(x(t)\).

c. Find an expression to determine how long it takes for the ball to reach its maximum height?

d. Assume that \(c / m=5 \mathrm{~s}^{-1}\). For \(v_{0}=5,10,15,20 \mathrm{~m} / \mathrm{s}\), plot the solution, \(x(t)\), versus the time.

e. From your plots and the expression in part \(\mathrm{c}\), determine the rise time. Do these answers agree?

f. What can you say about the time it takes for the ball to fall as compared to the rise time?