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

A ball is tossed up from a height h0 at time t=0 with an upward speed of v0. 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 h0, v0, G and ε. How high will the ball rise?

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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 v0, we must have
Integrating both sides of Eq.(1), we get

From the initial condition y(0)=h0, we must have
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 tmax, and the maximum height that the ball reaches as hmax, we have

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