Getting something for nothing?

From: numberdude (numberdude_at_netzero.net)
Date: 08/05/04 

Date: Wed, 4 Aug 2004 22:24:03 -0400

I was pondering the question, "Does it take the same amount of energy to
accelerate a mass from 10 to 20 mph (in say, one second) as it does to
accelerate it from 30 to 40mph (also in one second)?" In other words, for a
change in 10mph per second, will it always take the same amount of energy?

Kinetic energy = 1/2mv^2, so the change in KE = 1/2m(v2^2 - v1^2 )

For the case where v2 = 20mph and v1 = 10mph

  Change in KE = 1/2m(400 - 100) = 150m

For the case where v2 = 40mph and v1 = 30mph

  KE = 1/2m(1600 - 900) = 350m

It takes more than TWICE as much energy in the second case!

Now here's the puzzle: It takes a certain amount of force to produce the
change in velocity in both cases. Since the changes in velocity took place
in the same amount of time in both cases, the accelerations are the same.
Now we have equal masses moving with equal accelerations (but at different
velocities). Since the force needed to produce the acceleration is, F = ma,
the forces must be the same in both cases. But this means that equal forces
acting for equal times on equal masses will produce different changes in
kinetic energy at different velocities! It seems that at higher initial
velocities a given force will produce a greater change in kinetic energy.
But this is like getting something for nothing, which I know shouldn't
happen. Am I missing something?

Allen

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