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Name: David B.
Status: educator
Age: 50s
Location: N/A
Country: N/A
Date: 12/11/2004


Question:
In the kinetic energy formula, why do we divide by 2? My 6th graders want to know and I'm not really certain.


Replies:
David B.,

When we push on something in the direction of motion, we make it speed up. This adds energy, kinetic energy. The force we push with, times the distance we push, is called the work we do. Kinetic energy is set up so that this work equals the change in kinetic energy. It turns out that the one-half in the kinetic energy formula is necessary to make this happen.

When doing motion, there are four major constant-acceleration formulas. One that we see is v^2-v0^2=2ax. Remember, x is displacement. Multiply everything by mass: mv^2-mv0^2=2max. The net force on an object equals mass times acceleration: mv^2-mv0^2=2Fx. For constant acceleration work is W=Fx. Also divide by two: (1/2)mv^2-(1/2)mv0^2=W.

Ken Mellendorf
Math, Science, Engineering
Illinois Central College


Because otherwise energy would not be conserved!

To do the derivation quickly and roughly, starting with a mass m at rest with a constant force F acting on it. Then F = ma, or, if we multiply by x, Fx = max. Fx is just what is defined as the work done on the mass m by the force F acting over a distance x. The velocity of the mass after this acceleration is given by v^2 = 2ax. Remember the kinematics?

Then work = W = Fx = m v^2/2. And now it is clear that that pesky factor of 2 comes from the kinematics. You can get v^2 = 2ax from x = at^2/2 and v=at, which would be a good exercise to derive.

Best, Dick Plano, Professor of Physics emeritus, Rutgers U



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