Intuitively Understanding Work and Energy

Question

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 1

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.

Answer 2

After more digging, I came up with this quote from Feynman -

There is a fact, or if you wish, a law governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the conservation of energy.

It states that there is a certain quantity, which we call “energy,” that does not change in the manifold changes that nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says there is a numerical quantity which does not change when something happens.

It is not a description of a mechanism, or anything concrete; it is a strange fact that when we calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.

It is important to realize that in physics today, we have no knowledge of what energy “is.” We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. It is an abstract thing in that it does not tell us the mechanism or the reason for the various formulas.

As Manishearth's answer demonstrated, it is certainly possible to show the mathematical principles that go into understanding energy, but it seems to me to be a formula meant for mathematical convenience (as is Torricelli's equation), and not something meant to be intuitively understood in and of itself -

Generally it's better to look at [kinetic energy and momentum] as mathematical tools, and not attach them too much to our notion of motion to prevent such confusions.