PhysicsLAB: Derivation of the Kinematics Equations for Uniformly Accelerated Motion

PhysicsLAB

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Derivation of the Kinematics Equations for Uniformly Accelerated Motion

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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
v_f represents the object's final velocity at the end of the time interval
v_o represents the object's initial velocity at the beginning of the time interval

Equation #2: rearrange equation #1 for v_f

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

v_oforb₁
v_f 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	v_o	v_f	a	s	t

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