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)
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