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

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

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