How High? How Fast?

Date: 07/17/2002 at 13:12:51
From: Brian King
Subject: baseball falling to earth

Suppose you could throw a baseball straight up into the air at 100
m.p.h.  How high would the ball go on a wind free day?  And how fast 
would the ball be moving when it returned to earth?

Date: 07/17/2002 at 21:25:12
From: Doctor Ian
Subject: Re: baseball falling to earth

Hi Brian,

Do you want to ignore friction caused by air resistance?  For 
example, if you were on the moon, the ball would come back with the 
same speed it had when it left. 

Is that what you wanted to know? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/17/2002 at 22:39:53
From: Brian King
Subject: baseball falling to earth

I would like to know what would happen on earth without any wind 
blowing just gravity pull on the ball. If you were on the moon there 
is not enough gravity to pull the ball back to the moon so how could 
you test this theory?

Date: 07/18/2002 at 07:35:45
From: Doctor Ian
Subject: Re: baseball falling to earth

Hi Brian,

There's gravity on the moon.  It's just weaker than on earth, 
because the moon has less mass.  But wherever there is mass, 
there is gravity. 

(In fact, in one of the most famous experiments in the history of 
science, Henry Cavendish 'weighed the earth' by detecting the 
gravitational attraction between two lead balls suspended from 
very thin pieces of wire.)

Have you ever tried to run in water?  If so, you'll know that the 
water itself slows you down by creating a frictional force as you 
move against it.  Air provides the same kind of resistance to 
items moving through it, and the nature of the resistance is 
complicated - it depends in part on the speed of the object, 
which means you need either calculus or a computer (or a _lot_ 
of time and paper!) to solve problems where air resistance is 
taken into account. 

However, if there were no air on the earth, a ball thrown upward 
with a speed of S feet per second would be traveling at the same 
speed when it returned to the height from which it was thrown.  
The same would be true if the experiment were done on the moon - 
the only difference being that in between takeoff and landing, 
the ball would rise to a greater height on the moon. 

In both cases, the ball spends exactly half its time going up, 
and the other half coming down.  On the way up, it decelerates 
from its initial speed to a speed of zero; and on the way down, 
it accelerates from a speed of zero... over the same amount of 
time it took to decelerate, which brings it back up to the 
initial speed.  (It's sort of like this:  If you spent a dollar a 
minute until you ran out of money, and then collected a dollar a 
minute for the same amount of time it took you to spend what you 
originally had, how much money would you end up with?  Exactly 
what you started with, right?)

Does this make sense? 

The height the ball would reach would depend on the gravitational
acceleration at the surface of whatever body you were standing on. 
For earth, that is about 32 feet per second, per second.  (That is, if
you drop something, after one second it would be falling at about 32
feet per second; after two seconds, at 64 feet per second; and so on.) 

A ball thrown at 128 feet per second would come to a stop after 4
seconds, which would allow it to reach a height of 

  h = (1/2)(32 feet/sec^2)(4 sec)^2

    = 256 feet

100 miles per hour is about the same speed as 147 feet per second, so
this is in the ballpark.  And remember, this ignores air resistance. 

On the moon, the acceleration would be smaller, so the ball would take
longer to come to a stop.  But in both cases, upon returning to the
point from which it was thrown, it would be moving at its initial
speed. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/