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Galileo Galilei (1564-1642) | A Particle in Motion...Galileo's Life | The Parabolic Path...A Modern View
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The Parabolic Path...A Modern View
Vincent W. Lau

For this examination of the parabolic path, we will only be concerned with those objects which experience constant acceleration. Our discussion begins with velocity. Unlike speed, velocity carries a directional component with it. Thus, velocity is itself a vector and we will denote a vector here in boldface. If a bicycle travels a certain distance in a certain amount of time , we define the bicycle's average velocity as:

where and are the starting position and time respectively.

What happens when the velocity of the bicycle begins to increase or decrease? This is exactly the notion of acceleration - a change in velocity over a time interval. For a constant accleration:

Rearranging and taking for convenience, we have:

Returning for a moment to velocity, we again take and solve Eq.\ (1) for the position to obtain:

But the average velocity is just:

Taking this and substituting into Eq. (3) we now have:

Now we substitute Eq. (2) into this equation and get:

If is constant then, using the properties of vectors, we separate into its horizontal and vertical components, and find that they too are constant. We take the results from above and independently determine the x and y components of a particle's motion. Specifically, we use Eq. (5) and for an object that is thrown horizontally and experiences free fall (i. e. one that has an acceleration due to gravity , , and ):

We do the same for the velocity by using Eq. (2):

Plugging different t into the above equations and plotting the projectile's path from Eqs. (6) and (7) we will obtain a parabolic path. The explanation comes from examining Eqs. (8) and (9). Since the y component of the velocity is changed by the gravitational acceleration, the object will begin to move downward vertically. At the same time, the x component of the velocity experiences no change. Combine these two components of the acceleration and the object will continue to move horizontally while moving vertically until it hits the ground. Thus, a parabolic path is obtained.