# Projectile Motion

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Jugglers know that if you throw a ball, a bean bag, or a pin into the air, it will follow a curved path. This curve is what naturally happens when an object can move in two dimensions—horizontal and vertical—at the same time. As the ball moves horizontally, gravity pulls down. Physicists call this projectile motion. In this unit you will learn why any projectile, no matter if thrown, shot, or launched, will follow a curved path while in the air.

## Watch the Video

In the video, watch as jugglers Jake and Marty reveal the physics behind their act. Find out why even the world’s best jugglers can’t manage more than eight pins!

## Circus Physics: Projectile Motion

In this video students will learn why any projectile, no matter if thrown, shot, or launched, will follow a curved path while in the air.

### Questions to Consider While Watching the Video

- What determines how high a juggling pin goes?
- What determines how far it travels horizontally while in the air?
- How does the change in the pin’s vertical velocity compare to the change in horizontal velocity?

## Digging Deeper

Before we jump in, let’s agree on two things:

- Velocity is how much you change your position per change in time.
- Acceleration is how much you change your velocity per change in time.

In the diagram of the juggler, notice how the pin’s horizontal velocity—blue arrow—remains constant while the pin is in the air. The vertical velocity—red arrow—gets smaller as the pin nears the top of its curve. This is because the acceleration due to gravity—green arrow—is always downward. The changing vertical velocity, due to the force of gravity, is what causes the curve. Without the force of gravity the pin would travel in a straight line.

As the pin goes up, its vertical velocity slows down. By how much? For every second the pin stays in the air, its velocity goes down by 9.8 meters per second—the acceleration due to gravity. This means that if you tossed a pin into the air with an initial vertical velocity of 9.8 meters per second, it would take exactly one second for its vertical velocity to equal zero. This happens when the ball is at the very peak of its path.

After two seconds, the pin’s vertical velocity will again be 9.8 meters per second, but this time in the downward direction. If nothing stops the pin, after three seconds, it will be falling at 19.6 m/s, after four, 29.4 m/s, and so on.

Horizontally, there is no gravity, so there’s nothing to slow down the pin’s horizontal velocity. After one second, its horizontal velocity is exactly the same as it was when you tossed it…assuming there’s no air friction.

Note: Gravity only affects the vertical velocity, not the horizontal.

So, what determines how high the pin will go?

Gravity has to play a role. So does the initial vertical velocity—how hard you throw it. The final ingredient is the time the pin takes to reach the top of its arc. We can combine these to come up with a formula that tells you how high a pin will go:

H_{max} = V_{initial} x time + (½) x gravity x time²

So if we throw a pin with an initial velocity of 9.8 meters per second, we know it will take one second to reach the top of its arc. Its height at the top will be:

H_{max} = (9.8 m/s) x (1 s) + (½) x (-9.8 m/s/s) x (1 s)²

H_{max} = 4.9 m

Note that gravity is negative because it is acting in an opposite direction to the initial velocity.

## Your Turn

Use the concepts and formulas from this unit to figure out the following:

If you throw a juggling pin to a height of three meters straight up, how long will it take to come back to your hand?

##### Answer: 1.56 seconds

First, use the height equation to find the time it takes to fall 3 meters: 3 = ½ x (9.8) x t². Solving for t gives: t = √(2*3/9.8) ≈ 0.78 seconds. This is how long a pin takes to fall 3 meters, it will take the same amount of time to go up, so double this number, and you’ll have the total time of flight, ≈ 1.56 seconds.

## Further Explanation

If you’d like to learn more about projectile motion and juggling, check out these links: