Kinetic Energy

Kinetic energy is defined as the energy an object has through motion(specifically its velocity and mass). Mathematically, kinetic energy is defined as:

You can see a conceptual and mathematical derivation on the tree-extended article for the work-energy theorem. However, for now, let's just accept this equation given for the kinetic energy of an object.

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One thing you should remember is that kinetic energy is a scalar quantity and has to be nonnegative regardless because the mass of an object is never negative and the term where you square velocity will either be 0 or positive but never negative.

However, the change in kinetic energy of an object, given by

can be negative if the final velocity is lower in magnitude than the initial velocity. In fact, the original equation for instantaneous kinetic energy is implied from this second equation because usually, the initial velocity of an object is 0 m/s so the instantaneous kinetic energy of an object can be given by the first image.

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Intuitively, this should make sense with an example. Let's take an example of a hypothetical box on a floor with no friction. If I apply a force on that box over a distance, it'll speed up. If I keep applying that force and increase the displacement I apply it over, that speed will go up even more, and thus so will the kinetic energy of the box because we can take its mass to be constant(unless I somehow move it near the speed of light). Remember, this is all assuming that the force I apply on that object is the net force I'm applying to it.

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Now, let's look at this mathematically if the earlier intuition is confusing. The derivation for this theorem will also provide a derivation for the formula for kinetic energy.

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To prove the work-energy theorem mathematically, let's assume the object's acceleration is constant. You can come to the same formula through calculus but you'd be unable to assume that acceleration is constant so you'd have to integrate instead of just multiplying. Conceptually, this makes no fundamental difference so it's sufficient enough to use the simplified algebraic derivation.

We take one of our kinematic equations and solve for the product of acceleration and displacement. Multiplying in mass gives us net work, which is equal to half the mass times the difference in squares of the final and initial velocities of the object.

This derivation is known as the Work-Energy Theorem and it helps us properly connect the concept of mechanical work to kinetic energy.