momentum - Intuitively Understanding Work and Energy

It is easy to understand the concepts of momentum and impulse. The formula $mv$ is simple, and easy to reason about. It has an obvious symmetry to it.

The same cannot be said for kinetic energy, work, and potential energy. I understand that a lightweight object moving at very high speed is going to do more damage than a heavy object moving at a slower speed (their momenta being equal) because $E_k=\frac{1}{2}mv^2$, but why is that? Most explanations I have read use circular logic to derive this equation, implementing the formula $W=Fd$. Even Samlan Khan's videos on energy and work use circular definitions to explain these two terms. I have three key questions:

• What is a definition of energy that doesn't use this circular logic?


• How is kinetic energy different from momentum?


• Why does energy change according to $Fd$ and not $Ft$?

Answer

You may want to see Why does kinetic energy increase quadratically, not linearly, with speed? as well, it's quite related.

Mainly the answer to your questions is "it just is". Sort of.

What is a definition of energy that doesn't use this circular logic?

Let's look at Newton's second law: $\vec F=\frac{d\vec p}{dt}$. Multiplying(d0t product) both sides by $d\vec s$, we get $\vec F\cdot d\vec s=\frac{d\vec p}{dt}\cdot d\vec s $

$$\therefore \vec F\cdot d\vec s=\frac{d\vec s}{dt}\cdot d\vec p$$ $$\therefore \vec F\cdot d\vec s=m\vec v\cdot d\vec v$$ $$\therefore \int \vec F\cdot d\vec s=\int m\vec v\cdot d\vec v$$ $$\therefore \int\vec F\cdot d\vec s=\frac12 mv^2 +C$$

This is where you define the left hand side as work, and the right hand side (sans the C) as kinetic energy. So the logic seems circular, but the truth of it is that the two are defined simultaneously.

How is kinetic energy different from momentum?

It's just a different conserved quantity, that's all. Momentum is conserved as long as there are no external forces, kinetic energy is conserves as long as there is no work being done.

Generally it's better to look at these two as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.

Why does energy change according to $Fd$ and not $Ft$?

See answer to first question. "It just happens to be", is one way of looking at it.