PhysicsLAB Resource Lesson
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
  • 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
 
 
 
 
 
 



 
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