Throwing and Dropping Objects

Date: 11/26/2002 at 23:24:14
From: Amanda Evans
Subject: Math

I have tried asking my teachers this question, but they say I am too 
young to understand the answer. I have even asked my parents, but they 
give me the same answer. I cannot find this answer anywhere. My 
question is very simple.

Why, when you throw something in one direction and drop something at 
the same time, do they land on the ground at the same time? 

Also, when you throw something in the air, does it come down at the 
same pace as it went up?

Thank you for taking the time to read this.

Date: 11/27/2002 at 10:59:15
From: Doctor Ian
Subject: Re: Math

Hi Amanda,

It's a good question! A lot of people only understand this kind of
thing in terms of equations using variables, so they can't explain it
to someone who hasn't studied algebra yet. But let's see if we can 
skip the equations and just talk about the ideas, okay? 

The answer to your first question assumes that you throw the item in a
horizontal direction. Viewed from the side, the situation would look
sort of like this:

            throw
    '  '     ----->
    '     '
 d  '       '
 r  '        '    
 o  '         '
 p  '          '
    '           
    '           '
    '           '

If you throw the item in an upward direction, it will take more time 
to hit the ground. If you throw it in a downward direction, it will 
take less time. (By the end of this, you should understand why.) 

It might be easier to see what's going on if we imagine that you and I
are going to have an odd kind of race. We're both going to start at
the same point, and you're going to move south, following a simple
rule: In the first second, you'll take one step; then in the second 
second you'll take two steps; then three steps; then four steps; and 
so on:

     X
     |  1 step per second
    ---
     |  2 steps per second
     |
    ---
     |  3 steps per second
     |
     |
    ---
     |  4 steps per second
     |
     |
     |
    ---

What this means is that as time goes by, you'll be moving faster and
faster. It's like what a car does as it pulls away from a stop sign.
In the first second, it goes 5 feet; then 15 feet in the next second;
then 35 feet in the next second; and so on. This kind of motion is
called 'acceleration', which is just a fancy word for 'changing
speed'.  We say that the car 'accelerates' from a speed of zero to
whatever its final speed (say, 45 miles per hour) turns out to be. 

(When the car comes to the next stop sign, the speed decreases, and
sometimes we call that 'deceleration', but in both cases, it's really
an acceleration, that is, a change in speed.)

So in our race, you're accelerating toward the south.  

While you're heading directly south, I'm going to do something a 
little sillier. In the first second, I'll take one step east and one
step south; then I'll take another step east and two steps south;
then one and three; and so on. What would that look like? 

     X---
     |   |
     1   1---
     |       |
     |       |
     2       2---
     |           |
     |           |
     |           |
     3           3---
     |               |
     |               |
     |               |
     |               |
     4               4

So at the end of 4 seconds, I've moved to the east at a constant rate,
and you haven't; but we've _both_ moved toward the south, and with
the same acceleration. 

If we change 'south' and 'east' to 'down' and 'over', this is more or
less what happens when you drop one thing and throw a second thing in
a horizontal direction. 

Note that it doesn't matter how fast I move to the side. If I move
three steps east during each second, it looks like this:

     X---------
     |         |
     1         1---------
     |                   |
     |                   |
     2                   2---------
     |                             |
     |                             |
     |                             |
     3                             3---------
     |                                       |
     |                                       |
     |                                       |
     |                                       |
     4                                       4

So the curve that I make - or the curve made by the thing you throw -
has a flatter shape, but it doesn't affect the southward/downward
movement.  

Let's think about the car again. What makes it accelerate? In short,
we get the engine to burn fuel to produce energy, and we use the
energy to apply a force to the car - just as if we got out of the car
and started pushing on it from behind. (The engine can generate a lot
more force than a few people, though!) So whenever we apply a force to 
something, it accelerates. The more force we apply, the more
acceleration we get. (Sometimes in car commercials, you'll hear that a 
car can go 'from zero to 60 in ___ seconds'. If a car can accelerate 
quickly, it means that the engine can generate a lot of force.) 

When you drop a ball, it accelerates too. What makes it accelerate? 
As we currently understand it, everything in the universe exerts a
little bit of force on everything else, and we call this force 
gravity. So when you drop a ball, the earth is exerting a force on the 
ball in the same way that an engine exerts a force on a car. 

One difference is that when the car is going fast enough, we stop
pushing on the gas pedal, so it doesn't _keep_ accelerating. But the
earth never stops pulling. Even when the ball gets to the ground, the
earth doesn't stop pulling. When you step on a scale, and it tells
you that you weigh ___ pounds, what it's telling you is how hard the
earth is pulling on you. If the floor were suddenly to move out of the 
way (for example, if you were in an elevator that was standing still, 
and the cable broke), you'd start accelerating. The reason you _don't_ 
accelerate is that while the earth is pulling on you, the floor is 
pushing on you, and the forces exactly cancel out. It's sort of like 
if four people got out of a car, and two started pushing from the 
rear, and the other two started pushing from the front. They'd
generate a lot of force, but the car wouldn't go anywhere. And so we
have to amend our previous statement - whenever we apply a _net_ force
to something (that is, a force that isn't being canceled out by some
other force), it accelerates.  

So what is happening when you drop a ball? There is nothing to push
upward, so the downward force of gravity starts accelerating the ball
towards the earth. And it keeps accelerating until it hits the ground.  
Then a lot of other things happen that we're not going to talk about 
right now.  :^D

And what is happening when you throw a ball to the side? Again, 
nothing is pushing upward, so gravity starts accelerating the ball
toward the earth. The ball increases its speed in the downward
direction. What about toward the side? There are no forces acting in 
that direction, so it just keeps its original speed. 

So if you throw two balls, one twice as fast as the other, they'll hit
the ground at the same time, but one of them will travel twice as far
to the side. Dropping a ball, then, is just 'throwing' it with a
sideward speed of zero. 

There is one other important thing to know about gravity. If we put
the same engine in two different cars, one much lighter than the 
other, the lighter one will accelerate more quickly. In each case,
the engine generates the same force, but in the lighter car there is
less to push. 

But the force of gravity depends on how much stuff there is to pull
on. So let's say you have two balls, one of which weighs twice as much 
as the other. If you drop them at the same time, from the same height, 
the earth will exert twice as much force on the heavy one, but there 
is twice as much weight to be moved. So the extra weight cancels the 
extra force (sort of as if you bought a Porsche, but put half a ton of 
iron in the trunk), with the result that the two objects end up with 
the _same_ acceleration. That is, if you drop a baseball and a 
cannonball at the same time, from the same height, they'll hit the 
ground at the same time. In fact, if you go to a place where there 
isn't any air (like the moon), you can drop a feather and a piano, and 
they'll hit the ground at the same time. 

So what happens if you throw a ball upward? Now it's like a car with 
the brakes on. Gravity is pulling on it, so it decelerates until its 
speed gets all the way to zero:

       ^ 0 (stopped)
       ^ 1 'steps' per second
       |
       ^ 2 'steps' per second
       |
       |
       ^ 3 'steps' per second
       |
       |
       |
  start

But then it's just like dropping it from a new height.

       v 0
       | 1
       v
       | 2
       |
       v
       | 3
       |
       |
       v
  start

Since the acceleration is the same no matter which way the ball is
moving, when the ball gets back to its starting point, the speed
coming down is the same as the speed going up. If you're standing at
the edge of a cliff, and you throw the ball up at a certain speed,
when it gets to the bottom of the cliff it will be moving as fast as
if you'd thrown it _down_ with the same speed:

                  ^     
                ^   |
                |   v 
                ^   |
                |   |
                |   v 
                ^   |
                |   |
                |   |
                |   v
      ---------     | Now it's just as if you threw it downward.
              |     |
              |     |
              |     |
              |     v
              |     |
              |     |
              |     |
              |     |
              |     |
              |_____v_____
    

Does this all make sense?  

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

Date: 11/27/2002 at 18:49:27
From: Amanda Evans
Subject: Math

That all makes sence to me, but now I would like to know the equation 
for all of this. There must be a formula about my question.

Date: 11/27/2002 at 22:23:57
From: Doctor Ian
Subject: Re: Math

Hi Amanda,

In fact, there _is_ a formula for figuring out how far something falls
in a given amount of time. Here is an example of its use:

   Wile E. Coyote Lands in the River
   http://mathforum.org/library/drmath/view/56333.html 

And here is a slightly more interesting use:

   From What Height Did the Stone Fall?
   http://mathforum.org/library/drmath/view/56372.html 

And here is a more practical use:

   Real Life Uses of Quadratic Equations
   http://mathforum.org/library/drmath/view/60810.html 

Here is an explanation of where the formula comes from:

   Projectile Motion
   http://mathforum.org/library/drmath/view/56348.html 

It might be a little over your head, but I thought you'd like to see
it anyway.  

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

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