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Physics/divided by 2

Question
Greetings,

I've always wondered why kinetic energy's mv^2 is divided by two.

Below are two answers that I've gotten, could you please critique them and let me know anything true or false in them and which one is the right answer if either.  And if both are wrong what the real answer is.

Thanks,
Felicity

Dear Felicity,
For any constantly accelerated movement you can draw a graph of the change of velocity against the path traveled. At some arbitrary initial point the velocity change is zero. As the acceleration is constant, the velocity increases linearly with the path traveled, up to an arbitrary end point where the velocity change is maximal. This graph forms a right
angle triangle, and you can draw it along any infinitesimal part of the orbit. Now we should ask ourselves what is a constant velocity by which the object would travel the same path in the same amount of time. It is obvious that it is equal to 1/2 of the maximum. That is where the factor 1/2 comes from.

Hi Felicity,
When Gravesande dropped the lead ball into the soft clay, the ball started from rest and accelerated until it hit the clay.  That actual falling "Distance" that the ball travels is only "half" the distance of (Acceleration x Time^2).  Therefore that is the reason the kinetic energy equation of a falling object is divided by 2.  Hence 1/2 mv^2.  However with an orbiting planet this is not so.  The Velocity of the planet (Acceleration x Time) multiplied by the orbit Time is exactly comparable to the "Falling Distance" of Gravesande's ball.  The planet is not starting from rest therefore you would not divide the mv^2 in half.  The planet is traveling at "final velocity" throughout the orbit therefore if you divide the mv^2 in half, you are robbing it of half of it's true energy.  Therefore the kinetic energy of an orbiting planet = mv^2   not  1/2 mv^2

This is, unfortunately, going to involve some knowledge of derivatives or integrals (calculus).

The first one is wrong in the third sentence.  Velocity increases linearly with time under constant velocity and not path traveled.  But that person makes a good case for calculus...but then skips it.  The second one is closer, but not exactly right.  While properly referencing experimental data, they only provide a half-correct answer.

The true answer lies in the calculus that was invented to describe physics.  While energy is force multiplied by distance, the distance/time increases as the velocity increases.  One must integrate over the force*distance to get the total change in energy, which is the total change in kinetic energy when not dealing with friction, etc... (pure kinetic energy change).  Since you must integrate force (constant for constant acceleration and can therefore be pulled out of the integral) over distance (velocity*time) you must end up integrating over time for which the force is applied.  The integral, which you can look up in any math book on calculus, for a linear variable like time is t^2/2 and that's where the factor of 1/2 comes from, mathematically.

I kinda like the second argument.  If you think about it enough, it makes sense.  I still don't like the first because it's just wrong about velocity increasing linearly with path.  The velocity is quadratic with relation to the path, form the equation v^2=v_0^2+2a(d) where d is the path length.  That's clearly wrong to say that's linear.

P.S.  I hope this isn't a homework question, there's no way you could re-describe this to a teacher or professor...unless you're really good at calculus and take a long time to think about it.

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