3.5 Free fall  (Page 3/7)

Vertical motion of a baseball

A batter hits a baseball straight upward at home plate and the ball is caught 5.0 s after it is struck [link] . (a) What is the initial velocity of the ball? (b) What is the maximum height the ball reaches? (c) How long does it take to reach the maximum height? (d) What is the acceleration at the top of its path? (e) What is the velocity of the ball when it is caught? Assume the ball is hit and caught at the same location.

Strategy

Choose a coordinate system with a positive y -axis that is straight up and with an origin that is at the spot where the ball is hit and caught.

Solution

1. [link] gives
   
   $y=y_0+v_{0}t-\frac{1}{2}g{t}^2$
   
   
   $0=0+v_0(5.0\text{s})-\frac{1}{2}\left(9.8\text{m}\text{/}{\text{s}}^2\right){(5.0\text{s})}^2,$
   
   which gives $v_0=24.5\text{m/sec}$ .
2. At the maximum height, $v=0$ . With $v_0=24.5\text{m/s}$ , [link] gives
   
   $v^2=v_0^2-2g(y-y_0)$
   
   
   $0={(24.5\text{m/s})}^2-2(9.8{\text{m/s}}^2)(y-0)$
   
   or
   
   $y=30.6\text{m}\text{.}$
3. To find the time when $v=0$ , we use [link] :
   
   $v=v_0-gt$
   
   
   $0=24.5\text{m/s}-(9.8{\text{m/s}}^2)t.$
   
   This gives $t=2.5\text{s}$ . Since the ball rises for 2.5 s, the time to fall is 2.5 s.
4. The acceleration is 9.8 m/s ² everywhere, even when the velocity is zero at the top of the path. Although the velocity is zero at the top, it is changing at the rate of 9.8 m/s ² downward.
5. The velocity at $t=5.0\mathrm{s}$ can be determined with [link] :
   
   $\begin{array}{cc}\hfill v& =v_0-gt\hfill \\ & =24.5\text{m/s}-9.8{\text{m/s}}^2(5.0\text{s})\hfill \\ & =\mathrm{-24.5}\text{m/s}.\hfill \end{array}$

Significance

The ball returns with the speed it had when it left. This is a general property of free fall for any initial velocity. We used a single equation to go from throw to catch, and did not have to break the motion into two segments, upward and downward. We are used to thinking of the effect of gravity is to create free fall downward toward Earth. It is important to understand, as illustrated in this example, that objects moving upward away from Earth are also in a state of free fall.

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