Full Proof of the WorkEnergy Theorem
Though a calculus based proof of the WorkEnergy theorem is not completely necessary for the
comprehension of our material, it allows us to both work with calculus in a physics context, and
to gain a greater understanding of exactly how the WorkEnergy Theorem works.
Using that equation, the equation we derived for work done by a variable force, we can
manipulate it to yield the workenergy theorem. First we must manipulate our expression for the
force acting on a given object:
F
net
=
ma
=
m
=
m
=
mv
Now we plug in our expression for force into our work equation:
W
net
=
F
net
dx
=
mv
dx
=
mvdv
Integrating from
v
o
to
v
f
:
W
net
=
mvdv
=
mv
f
2

mv
o
2
This result is precisely the WorkEnergy theorem. Since we have proven it with calculus, this
theorem holds for constant and nonconstant forces alike. As such, it is a powerful and universal
equation which, in conjunction with our study of energy in
the next topic
, will yield powerful
results.
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 Fall '10
 DavidJudd
 Physics, Energy, Force, Work, WorkEnergy Theorem, Equilibrium point

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