This derivation is based on the properties of a velocitytime 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 (
t_{2}  t_{1}) 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_{o }for_{ }b_{1}

v_{f} for b_{2}

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 






























