Principles of Projectile Motion
Projectile motion involves objects that are only under the influence of gravity. Freefall, which you’ve studied, is a type of projectile motion that only involves objects moving vertically. But how do you deal with situations involving both horizontal and vertical motion?
In order to simplify this topic, it helps to perform a thought experiment originally presented by Galileo. If a ball is dropped from the top of the mast of a sailboat that is moving at a constant velocity, where will it land on the boat? If you think about this from the point of view of a person on the shore, you might think that the moving boat would move right out from under the ball, leaving it to fall towards the back of the boat. However, if you think about being a person on the boat, it’s as if you’re standing still, and the sea is moving below you. From this perspective, there’s no reason that objects dropped or tossed “straight up” should do anything other than fall straight back down.
Your experience should also point to this same conclusion. If you’re in an airplane that is moving at five hundred miles per hour and drop something off your tray, it doesn’t suddenly come crashing back into your stomach at five hundred miles per hour. From your perspective, it falls straight down.
Take a look at this animation from the Department of Physics at the University of Toronto to compare the motion of the dropped ball on the sailboat and see if it matches your expectations.
Relative motion is defined in terms of your frame of reference. It turns out that all motion can be thought of this way. If your frame of reference is not accelerating, it is called an “inertial reference frame”, and any physics experiment you perform should end up with precisely the same results as if you performed it in any other “inertial reference frame”.
The reference frame you are most familiar with is the one that moves along with the surface of the earth. You don’t feel like you’re moving, and you don’t have to account for the motion of the spinning earth when performing your lab experiments. In reality, you are moving somewhere near one thousand miles an hour as you ride the surface of the earth around its axis every day. The entire solar system is currently circling the center of the Milky Way galaxy at nearly half a million miles per hour, and our galaxy is also hurtling through space relative to our galactic neighbors. However, you do not need to include any of these motions when analyzing physics experiments because the results will be the same as those determined at rest.
But, what is important is that the vertical motion of freefall is independent of any horizontal motion. You can use this fact to easily analyze projectile motion by splitting it into its horizontal and vertical components and analyzing each separately.
For another example, take a look at what is called a “ballistics cart”. When this simple device is at rest, it will shoot a ball straight up and catch it again. But what can you expect when the cart is moving at a constant velocity? Take a look at this animation from the Boston University Physics Department, which illustrates a cart moving at a constant velocity that shoots a ball upward just before entering a tunnel. Where will the ball land?
If you show the animation with “no tunnel” and also show vx, you can see that the ball and cart are both moving at the same initial velocity. When the ball is shot upwards, the horizontal motion doesn’t change. The ball remains precisely above the cart, so when it returns to the cart, it lands right in it.
From the reference frame of the cart, the ball is not moving horizontally. It moves straight up and straight back down again. But from the reference frame of the ground, the ball follows another path. The vertical motion and horizontal motion are independent of one another and can be analyzed independently.
Finally, look at an animation of two spheres, one dropped, and one projected horizontally from the same height at the same time. This animation is also from the Boston University Physics Department. Which ball will hit the ground first?
When you play the animation, you’ll see that they both hit the ground at precisely the same time.
But you should also take a look at the position of each sphere as it falls. The locations are marked at equal time periods. You should notice that the blue sphere moves to the right precisely one grid space between marks. It has a constant horizontal motion.
You should also take a look to see that the distance that each sphere falls is the same at each marked time. The spheres fall a greater distance during each time interval for later times because the vertical velocity increases. However, you should notice that the vertical positions of the spheres are the same at each time during the motion. You can step forward and backward through the animation to see that this is the case.
For a closer look, select “blue ball” on the animation. Here the horizontal and vertical motion are separated. On the top, the simulation shows constant horizontal motion. On the left, the simulation shows freefall motion. In between, the simulation shows the actual position of the sphere as time passes. Again, you should line up the actual position with both the horizontal and vertical positions for each time shown by stepping backwards and forwards through the animation.
Why does the vertical position change more during each subsequent time period?
The ball is accelerating downward due to gravity at a rate of nine point eight meters per second per second. Since the downward velocity is increasing as time passes, the distance traveled in equal time periods will continue be larger.
Why is the change in horizontal position constant for each time period?
The ball is not accelerating horizontally. Since the ball moves with constant velocity, equal distances are traveled in equal time periods.
Projectiles: Equations and Variables
Now that you have looked at motion from several points of view, you will learn how to recognize the horizontal and vertical components of the motion. Click the player button to begin.
View a printable version of this interactivity.