Resource Lesson Derivation of the Kinematics Equations for Uniformly Accelerated Motion
This derivation is based on the properties of a velocity-time graph for uniformly accelerated motion where the
• slope of the graph represents the acceleration
• graph's area represents the displacement

Equation #1: slope = acceleration

Starting with the slope

where

gives us our first equation:

In this equation

• a represents the object's uniform acceleration
• t represents the interval of time ( t2 - t1) over which the object's velocity changed
• vf represents the object's final velocity at the end of the time interval
• vo represents the object's initial velocity at the beginning of the time interval

Equation #2: rearrange equation #1 for vf

Equation #3: area = displacement

Before we use the variables from our graph, let's take a moment and remember from geometry the formula for the area of a trapezoid.

On our graph, this trapezoid is turned over on its side and looks like

Substituting in the following variables

• vo for  b1
• vf  for b2
• h for t

allows us to rewrite the area of a trapezoid as kinematics equation #3

Equation #4: multiply equation #1 by equation #3

Equation #1:
Equation #3:

Equation #5: substitute equation #2 into equation #3

Equation #2:
Equation #3:

EQUATION SUMMARY (these MUST be memorized)

 Equation vo vf a s t