Deriving the Work Energy Theorem

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The Work Energy Theorem and Newton’s 2nd Law

The Work Energy Theorem can be derived mathematically in several ways. One method was shown by physicist James Clerk Maxwell in his 1877 book, “Matter and Motion”. A few physics instructors use calculus while many others take a far simpler approach. The method shown here is as simple as it gets; it takes the case of an object that begins at rest and uses the idea that force is a constant throughout.

We begin by writing Newton's Second Law in its usual mathematical form.

F=ma

Next, we factor time (t) into both sides of the equation.

Ft=mat

Since the velocity (v) of an object undergoing a uniform acceleration is related to the product of time and acceleration, we can then modify the preceeding equation to produce the simplest form of the impulse (Ft) momentum (mv) equation.

Ft= mv

Naturally, this equation is only valid for objects that begin at rest.

Note; often an instructor begins with the impulse / momentum equation because the lessons on them have usually already taken place.  From basic mathematics, it is a simple matter to determine the distance an object travels during a period of uniform acceleration. If an object begins at rest, its initial velocity is zero. And so by dividing the object’s final velocity (v) in half, we find the object’s average velocity. The time an object travels at an average velocity gives us the total distance / displacement (d) it has traveled.

distance 

Multiply both sides of the impulse / momentum equation by the one for distance using the corresponding members as shown below.

deriving work energy

We now divide by the time (t) force acts on both sides of the equation.  This produces the simplest form of the Work Energy Theorem that works whenever an object begins at rest.

we

Other versions exist and they take into account such things as the direction of force and objects that begin at velocities other than zero. The addition of the “cos ϴ" factor exists to account for those situations where the angle of force and the displacement are not full in accord.

In terms of the Scientific Method, the fact that the Work Energy Theorem can be derived mathematically is scientifically irrelevant. This is because this theorem describes a physics principle, not a mathematical one and so, it must be tested within a physics context. In other words, the Work Energy Theorem describes a phenomenon (mechanical energy) that is different than the physics principle (Newton’s 2nd Law) and the basics mathematics upon which this theorem was derived.

 

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