# Uniform Acceleration

If an object has a constant acceleration vector, we say it undergoes uniform acceleration. Most MCAT problems involving acceleration will involve uniform acceleration. For uniform acceleration, we have the following:

(5)

that is, the average velocity over a period of time is the average of the beginning and ending velocities. This may seem like a natural definition of average velocity, but the definition of vavg is given by equation (2), and equation (5) holds only for uniform acceleration.

See Figures 2-7 and 2-8 for an example. The velocity v1 is small, v2 is large, and vavg is exactly between them.

If we start with the definition of average velocity, we can write

Î”x = vavgÎ”t

This is a useful equation if you do not have and do not need the acceleration (see equation [7] below). Furthermore, if we substitute v2 = v1 + aÎ”t (from equation [4]), then we obtain

......(6)

This is the first equation which may seem a bit arcane. You should memorize it anyway. Working through the algebra will help you memorize it.

Example

A car is accelerating uniformly from rest. If it goes a distance d in the first second, how far will it go in the first four seconds?

Solution

We want an equation involving the quantities mentioned in the problem, av1 = 0, Î”x, and Î”t, so equation (6) is it. With v1 = 0, we obtain

If Î”t increases by a factor of 4, the Î”x increases, and it increases by a factor of 42 = 16.