Why is it that kinetic energy is equal to a half times mass times velocity squared? If I ask my students to hazard a guess at what they think the kinetic energy formula should be, they often think that it's related to mass and related to velocity, and might hazard a guess at mass multiplied by velocity. But this formula is wrong.
It's not mass times by velocity, it's actually a half times mass times velocity squared. But why is it... that we have a term in a half and why is it that it's velocity squared?
Well to answer that question imagine a car that's accelerated along a road by a force F. Now if that force is the only force acting on the car then it should move with uniform acceleration, that speed up uniformly. But that's not always the case with cars because of friction and air resistance. So instead of a car I'm going to ask you to imagine a spaceship.
Now the spaceship in deep space has no air resistance, and so when you apply a force to it, it really will accelerate uniformly. And the force needed to produce an acceleration is related by the formula F equals ma. The force provides an acceleration a on the spaceship of mass m. This would be the velocity-time graph you'd get, and let's say that in a time t, the spaceship reaches a final speed of v. So the acceleration of the spaceship...
is V divided by T, change in speed over time for the change. Now in that time the spaceship will travel a distance given by the area under the graph or distance is a half times V times T. Now we want to work out the energy gained by the spaceship and the energy will be gained will be done work done by the force. Work done is force multiplied by distance And so the kinetic energy of the spaceship, after it's been accelerated by the force, will be F multiplied by D. Now we can simplify that equation further.
We know kinetic energy is force times distance, but over here we decided that force was equal to M multiplied by A. So we can replace MA where we had force. Similarly, we had distance here, and we can replace distance with a half VT.
And so we get the formula kinetic energy equals ma half vt. Again, that formula can be simplified further. We know that the acceleration is equal to change in speed over time for the change. So we can replace va with v over t. You'll notice that there are two t's and they should cancel.
So cancelling the t's will give us... this equation. Now you'll notice there are two v's and v multiplied by v makes v squared. And so now we've almost got our kinetic energy formula.
It's just a matter of rearranging the m and the half and you get a half mv squared. And so that's why kinetic energy is equal to a half mv squared.