Equations of Motion

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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

Eq of Motion 1.png

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

Eq of Motion 2.png

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:




which leads to:


This can finally be written in the form:



Free Fall

Projectile Motion