Derivation of the kinetic energy equation. Where
did the KE equation come from? Let's see.
Let's say there's an object of mass m, at rest. In order to accelerate
it to some speed v, we are going to exert a force on it. Let's say the
force we exert is the only force, so is equal to the net force
on the object. Now, let's calculate the work we do.
Newton's second law of motion tells us that FNET can be replaced
by m·a. Also the distance, d, is equal to the average velocity times the
time interval. If you're starting from rest, the average velocity is half the final velocity, v. This gives us...
Now, recall that the acceleration, a, is equal to Δv/Δt,
so let's substitute that in...
The Δt's cancel. Also, because we started
from rest and ended with velocity v, then the change in velocity, Δv,
is equal to the final velocity v.
And the final simplification yields
This, then, is the work done in accelerating a mass m from rest to speed
v. The direction of the velocity doesn't matter; only its magnitude (speed)
does. We define the kinetic energy to be equal to this quantity.
Notice also that KE is always positive.
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