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Projectile Motion

It is important to note that projectile motion is simply a special case when it comes to motion. It is motion in two dimensions – over and down – horizontal and vertical. We will not use new equations or anything else new, we will only use the equations we already know and use them in this special case.

As I said, projectile motion consists of two components, horizontal and vertical, that are independent of one another. That means that if you drop a ball and throw another one straight out, they will both hit the ground at the same time. This is not something that is obvious but if you think about the ball that drops straight down gets pulled down by gravity and the ball that is thrown straight out gets pulled down by gravity as well. Both vertical motions are identical. The ball that gets thrown has another component added however, and that is that it is moving out, or moving horizontally in addition to falling. Horizontal means “parallel to the horizon”.

Check all the way at the bottom of the page for another picture.

 

Let’s look at the equations. The horizontal motion is uniform because nothing speeds up or slows down the ball that is thrown. The only equation that we have for horizontal motion is eq 1. The vertical motion is essentially free falling motion. It is determined by gravity and we have two equations: eq 2 and eq 3 where a is gravity (10m/s/s). It is important to realize that the only thing these two have in common is time. Even though there are two d’s in the equations they mean something different, one is a distance the other is a height.

When solving a problem it is best to make a table with two columns, one for the horizontal the other for the vertical component. Put the equations into the respective columns and decide into which column each number given belongs and what it is you are looking for.

 

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Here is an example: A tennis ball is hit and leaves the racket with horizontal speed of 20 m/s. The ball hits the court at a horizontal distance of 10 m from the racket. What is the height of the tennis ball when it leaves the racket?

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Once you have the table fill in the numbers. The first number is a velocity, it is the velocity with which the ball leaves the racket. Try to picture this and you will see that it is the horizontal velocity, therefore it goes in the left column. The distance of 20 m is designated “horizontal” so there is no question where this goes – into the left column again. You are looking for the height of the racket, which is the vertical component. We have the equation eq 4 where h (height) was substituted in for d. We have the gravity but we do not have the time. So we must first solve for the time in the left column (horizontal). The equation was eq 5 where v = 10 m/s and d = 20 m. The time is therefore 0.5s, not 2s as you might get for answer solving this equation wrong. Remember, if the ball goes 20 m in 1 second it cannot take 2 seconds to go a smaller distance. If the distance is half, so has to be the time – 0.5 s. This was proportional reasoning. The bal is in the air for half a second and therefore drops for half a second. We plug 0.5 s into the equation eq 6 and we get for the height 1.25 meters.

 

The picture below shows he independence of the two components. Three balls are moving at the same time. The blue ball is dropping, the red ball is projected to the right and the green ball is moving at constant velcoity on the ground. The red ball will hit the green ball all the way at the right at the moment when the blue ball hits the ground. The vertical motion is only governed by gravity, that means that both the red and the blue ball hit the ground at the same time.

 

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