**Problem 1A-9**:** Vertically-Thrown Ball with Air Resistance

A ball is tossed up from a height h_{0} at time t=0 with an upward
speed of v_{0}. The vertical position of the ball is expressed as a
function of time t, y(t), and satisfies the equation of motion

where G and ε are positive constants. Obtain y(t) and the speed of the ball v(t) in terms of h

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**Solution:**

The equation of motion of this problem is the same as the equation of motion of the previous proble, Problem 1A-8, so the general solution is also the same. The general solution is

Since the initial speed of the ball at t=0 is v

Thus

Integrating both sides of Eq.(1), we get

From the initial condition y(0)=h

Thus

Eqs.(2) and (1) determine y(t) and v(t) completely in terms of given constants. To find the maximum height that the ball will rise to, we observe that v(t)=0 at the time when the ball rises to its maximum height. We can find this time t by solving v(t)=0, thus from Eq.(1) we get

Call this time t

Thus

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