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September 19th, 2013, 03:50 PM   #1
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find the average speed of the ball

the height in ft of a baseball, t seconds after being thrown straight upward, is given by h(t)=36t - 16t^2

1) find the average speed of the ball for
a) 0? t ?1
b) .5? t ? 1
c) .9 ? t ?1

the answer to a) is 20 ft/sec, which is what you get if you plug 1 into the equation. the answer to b) is 12 ft/sec, and c) is 5.6 ft/sec.

how the heck did they get these answers?? I just don't get it. please explain how to do this step by step

also it asks to estimate the speed of the ball after 1 second
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September 19th, 2013, 05:09 PM   #2
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Re: find the average speed of the ball

Hi coolbeans33, and welcome to the forums.

Differentiate h(t) to find v(t), the speed* of the ball. You will see that the equation is linear in t, so to find the average, you can simply add the values at the endpoints and divide by 2. You should get the answers that you stated.

As far as the estimate is concerned, gravity reduces the speed of the ball by 32 ft/s each second. So after 1 second, subtract 32 ft/s from the initial speed to find the speed of the ball. But this is exactly how you would calculate it using the equation for v(t). I guess the problem says estimate since the 32 ft/s is not very precise.

* The derivative of position is technically the ball's velocity rather than speed, meaning that it has direction, but in this case we can ignore direction and take the calculated value as the ball's speed.
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September 21st, 2013, 08:00 AM   #3
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Re: find the average speed of the ball

This looks to me like an "introduction to the derivative" type problem so differentiation might not be appropriate. (And, it works in this case, as jks said, only because the velocity function is linear.) In any case it is much simpler to use the basic definition of "average speed".

To find the average speed, in ft/sec, find the distance it has moved and divide by the time required:
The average speed between t= 0 and t= 1 is . The average speed between t= 0 and t= 0.5 is , etc.
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