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 (
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
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