Projectile Motion
a.k.a - How to hit your neighbors with a cannon ball

If you only read one thing read this!
Please don't read only this, I wrote a whole lot more :-)

Here is the deal, the most important thing to remember about projectile motion is this : THE VERTICAL AND HORIZONTAL MOTION OF AN OBJECT DO NOT AFFECT EACH OTHER! Sorry for yelling, but that was very important.

Now the details...

As a quick review, vertical is just a smart way of saying up and down, and horizontal is just a smart way of saying left and right. So what do I mean that the horizontal and vertical motion do not affect each other? OK lets go.

I'm standing on the top of a stationary bus with you. We drop a ping pong ball on the table on the bus. Why is there a table on the bus? Because I am the writer of this little passage and I can make things however I want. Anyway... What do we see? The ball accelerates downward (falls), hits the table and bounces back up. Fascinating. We can both agree that the ball only moved vertically. I.E. it only moved up and down. The ball didn't come any closer to you or me. It went straight down and then straight up. So far so good.

The Vertical Motion of the Ping Pong Ball

Now lets suppose we have a common friend. I know that's improbable, but once again, I'm the writer, so deal with it. Lets have the third friend stand out side the bus. Once again we will drop the ball, but this time lets have the bus driving past the friend as we do so. Now things are slightly different. You and I and the ball are all moving at the same speed, as long as we are not accelerating (the bus is moving at a constant speed) in principal we cannot tell that we are moving at all. Yes I know, we can see trees, buildings, and Barney the purple dinosaur moving past us, so we know that we are moving, but from a physical standpoint, we can't prove that we are moving. After all, we could be standing still and everything else could be moving past us. An interesting thought, but I digress.

On the bus we see that the ball only moves up and down, but off the bus, we see the same ball move up and down and to the right

OK, back to the point. Our mutual friend sees the ball moving in a arched path, shown above. We however see the ball move exactly as it did before, that is to say, only up and down. So we, and our friend are seeing the same ball moving in different directions. We disagree on the path that the ball takes. This whole story was supposed to show that the horizontal and vertical motion are independent. If they were not the horizontal motion of the bus would have changed how you and I saw the ball move vertically. No matter how fast the bus is moving to the left, the up and down motion is going to be affected and no matter how high we throw the ball, the horizontal motion will not be affected. The only thing that is the same about the two motion is the time. The time up is the same as the time across. This turns out to be the fact that lets you solve all the problems.

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Now it gets messy
Reducing the motion to one dimension. Yes, it is as boring as it sounds.

The Initial Velocity of our Projectile Baby   OK, because we can reduce the motion to one direction, we will. Trust me, it makes the problem simpler, in a more complicated sort of way. In order to do this we must sell out to the daemon of trigonometry. Lets say we have a projectile, a human infant, just to make it fun. Lets throw him with a velocity of 100m/s, 33º off the ground. How fast is the baby moving horizontally? To know that, we must use the COS function.

COS 33º=.84
(.84)(100m/s)=84m/s

The Horizontal Motion of our Projectile Baby

 

So we know that the baby is moving at a speed of 84m/s horizontally. We also know since the baby is airborne, that there are no forces acting on him in the horizontal direction. Therefore, the baby will always be traveling at 84m/s to the right as long as he is in the air. What about gravity! What about it? I said in the horizontal direction. If you remember, gravity only pulls down, so it has no affect on the baby's horizontal motion. Gravity only affects the vertical motion. And speaking of the vertical, lets find the baby's vertical speed. To do that, we must use the SIN function.

SIN 33º=.54
(.54)(100m/s)=54m/s

The Vertical motion of our Projectile Baby

OK, so we know that the baby is moving with a constant speed of 54m/s vertically right? WRONG! That statement implies that there is no gravity! One key to being good at physics is to constantly visualize what is going on. The 54m/s is the baby's initial velocity. Gravity slows him down.

  The Vertical Component of the Baby's Projectile Motion

Solving the problem
Reach for the sky young lad!

If you want to solve projectile motion problems, more often than not, you have to look at the vertical motion. That's where the good stuff happens. Although the horizontal and vertical motion are independent of each other (one does not affect the other), the time will always be the same. What the hell am I talking about? Well lets make this more concrete and solve some problems. Suppose you wanted to know the horizontal distance the baby travels. Well what do we know? The baby does not have any forces acting on him in the horizontal direction, so he is not accelerating. So the only formula we can use to describe the horizontal motion is v=s/t. Since we want to solve for the displacement, lets rewrite that as s=vt. We know the velocity is 84m/s but we do not know the time. So lets do something about that.

What determines the time? The vertical motion determines how long that baby will be airborne. Think about it, the baby is going to stop moving when he smashes into the ground. So how are we going to find the time. There are a couple of ways to do it, but here is the simplest.

The Horizontal Motion Stays The Same While The Vertical Motion Constantally Changes.

Start out with what you know about the baby's vertical motion. Well, we know the initial speed. So we know the viy=54m/s.

What is that sub y for? Well in projectile motion since we are dealing with horizontal and vertical motion at the same time things can get confusing with all the different velocities, so to keep things straight we usually write a sub y and sub x along with the velocities to distinguish between them. viy just means, the initial velocity in the vertical direction. vfx would mean the final velocity in the horizontal direction.

OK, what else do we know about the baby's vertical motion. Well the baby is flying away from the surface of the earth, so what force would affect his motion? That's right! Gravity affects his motion. So we know that ay=-9.8 m/s2. Why negative? Well if the baby's velocity is positive and he is slowing down, that means he is accelerating in a direction opposite to his motion. Most people can't see that right away, so here is another way to think of it. If the baby's acceleration and velocity were positive, he would just keep going up forever, or at least until he reached outer space and exploded. So if you want to keep that baby on earth the acceleration must be negative.

Enough babbling, lets solve the problem. What else do we know? Here is the trickiest part. We also know that when the baby reaches his maximum height his velocity will be zero, but only for an instant! As the baby travels upward he keeps slowing down, slowing down, slowing down, until he reverses direction (starts to fall toward Earth). In order to reverse direction he must stop moving for an instant. So if we are just looking at the baby's upward trip we would say that vfy=0 m/s. Believe it or not but we now have enough information to solve for how far horizontally the baby travels!

First we want to find the time the baby is airborne
Using the formulas from the kinematics section we know that...

vf=vi + at

Remembering that we are doing projectile motion we rewrite that using 'y's to show that we are talking about vertical motion

vfy=viy + ayt

Now substitute in what we know.

0m/s=54m/s + (-9.8m/s2)(t)
Solve for t
(-54m/s) / (-9.8m/s2)=t
t=5.5s

DO NOT FORGET! THAT WAS THE TIME IT TOOK FOR THE BABY TO REACH HIS MAXIMUM HEIGHT! THE TOTAL AIR TIME OF THE BABY IS TWICE THAT NUMBER! It takes just as long for the baby to fall back to the ground as it did for him to reach his maximum height.
The total air time=11s

Whew! We are done... Wait a second! What the hell! That has nothing to do with the range! I hear you all cursing me under your breath, but trust me this is actually going somewhere. No really it is. We want to find the horizontal distance traveled right? Since the horizontal motion does not undergo acceleration, we can use the formula vx=s/t. Rewritten to solve for s we have s=vxt. Before we could not solve this because we didn't know the time. But now we do!

s=vxt
s=(84m/s)(11s)
s=924m

So after all that we know that the baby traveled 924m horizontally before he crashed into the ground. Don't you feel all satisfied?

If any of the above was unclear, or if you have any comments or suggestions, please E-mail me!

ColinGPalmer@hotmail.com





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