# Equations of Motion

*Helena Dedic*

## Derivation of Equations of Motion

Consider a v-t graph for an object moving with constant velocity.

• Motion with constant velocity implies that the acceleration is equal to zero and therefore the v - t graph is a horizontal line.

• The displacement is the area under the v - t graph

We write

• Motion with constant acceleration implies that the velocity is a linear function of time. Given that the initial velocity is we can write

where the acceleration is the slope of the graph and is the intercept.

• The displacement is the area under the v - t graph

We write

• The displacement in both cases is

• There are two independent equations describing motion with constant acceleration and five variables: , v(t), , a and t. Consequently, in any problem we can solve for two of those variables and three other must be given.

• We can derive another useful equation by eliminating t from the two equations above.

We begin with equation

We isolate t:

Then we substitute for t in the equation for the displacement:

We will now multiply both sides by 2a. This step will eliminate fractions from this equation.

This leads to:

Expanding:

which leads to:

This can finally be written in the form: