# How Acceleration Relates to Kinetic Energy

Q:
When a mass is accelerated, where does come from its kinetic energy? Does it come from the force applied to the mass in order to accelerate it? If so, is there any mean to connect newton with joules i.e. 1N of force increment a 1kg mass' kinetic energy by X joules/sec?
- Anonymous
A:
How a mass accelerates is an interesting question.  When a force is applied to an object, that object's momentum changes as well as (sometimes, as explained below) its kinetic energy.  For this question, I'll focus mainly on how a force affects kinetic energy.

At low speeds and energies, all of the forces acting on an object equal that object's mass times its acceleration (called Newton's 2nd law).  The relationship between force and energy can be derived from the aforementioned 2nd law:

So we have first F=ma (Newton's 2nd law) where F is force, m is mass, and a is acceleration.  It should be noted that any bold letters are vectors meaning that they have magnitude and direction.

This can be rewritten more generally as F=(dp/dt) where p is momentum and dp/dt implies a change in momentum with respect to a change in time.  Momentum, p, however, is related to kinetic energy, KE, by the equation KE=p2/2m.  So a change in momentum corresponds to a change in kinetic energy.

This is the essence of Newton's second law: Applying a force to a mass changes the  momentum of that mass.  An acceleration just represents this change in momentum for an object that has a constant mass.

The units newtons and joules can be connected directly.  For a mass under a constant force, F.Δx=ΔKE where the Δ is the symbol for change and the "." means that only the part of F that is in the same direction as x should be multiplied.  So moving an object a distance x with a force F changes the kinetic energy in a mathematically direct fashion.

This description of forces, masses, and energies is a little simplistic.  and have more information on derivations of kinetic energy and how it relates to force.  Wikipedia's article has some mathematical rigor but both are very informative.

Mark

(published on 07/05/2010)

## Follow-Up #1: Kinetic Energy is Relative

Q:
Kinetic energy is entirely relative. Picture this: imagine a very long platform rolling on hundreds of wheels. The platform moves eastwards at a constant 10m/s. On the platform there is a 1000kg car that moves in the opposite direction (westwards) at a constant 10m/s. For a platform-based observer, the car has 50kJ of kinetic energy (0.5*1000*10^2) since it moves towards/away from him at 10m/s. However, for a ground-based observer, the car moves at a constant 0m/s since to him the platform and the car move at opposite directions at the same velocity. Thus the car has no kinetic energy (0.5*1000*0^2). If the driver were to hit the brakes a very funny thing happens. To the platform-based observer the car decelerates and stops: kinetic energy is transformed into heat (brakes). But for the ground-based observer the exact opposite happens! The car accelerates in the direction of the moving platform and some of the kinetic energy of the platform is transfered to the car. That transfer of energy creates heat and heats up the brakes. So kinetic energy (along with maaany other things) is reference-frame dependent. Only a car moving exactly at c would not be reference-frame dependent, but moving at c is impossible. In that sense, kinetic energy is not something that has an exclusive physical existence (as if the car kinetic energy is the same regardless of the reference frame). Considering this, we can hardly say that a force is converted to kinetic energy. This reminds me of a funny sentence: Is the motion of westwards moving airplane that moves at the same speed of the Earth's rotation according to the airplane's latitude is due to its motors pushing out compressed air or is due to the fact the Earth is really spinning on itself under the airplane? Both answers are correct! Boy, how relativity is awesome!
- Anonymous
A:
You're certainly right that kinetic energy is relative, depending on the choice of reference frame. That's part of Newtonian physics, although the form of the dependence on reference frame is different in modern relativity.

Looking at the expression for the effect of force on kinetic energy, even in the Newtonian approximation in which F is the same in any inertial frame, obviously Δx isn't. Thus the effect of a force on KE is entirely dependent on choice of reference frames, as you say.

As for the airplane, the important thing is not the arbitrary choice of reference frames (e.g. at rest with respect to the center of the earth or at rest with respect to the nearby surface of the rotating Earth) but rather that the plane is kept up by the wings thanks to the relative motion of the plane and the air. That relative motion is caused by the engines, not by the Earth's rotation.

Mike W.

(published on 04/07/2011)

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