# 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

See Figures 2-7 and 2-8 for an example. The velocity

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

*v*_{avg}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

*v*_{1}is small,*v*_{2}is large, and v_{avg}is exactly between them.If we start with the definition of average velocity, we can write

Î”*x* = *v*_{avg}Î”*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

*v*_{2}=*v*_{1}+*a*Î”*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, *a*, *v*_{1} = 0, Î”*x*, and Î”*t*, so equation (6) is it. With *v*_{1} = 0, we obtain

If Î”*t* increases by a factor of 4, the Î”*x* increases, and it increases by a factor of 4^{2} = 16.