# How do we get definitions of fundamental terms from a formula like the Work-Energy Theorem?

I’m a senior taking AP Physics C, and I’m currently losing my mind over this question.

I get that work is essentially a measure of whether a force is “successful” or not in displacing an object in its direction. And, that if said force is successful, the velocity of the object must change in response and gain kinetic energy. I think that kinetic energy is just the ability to influence another object’s acceleration by motion. But here’s the thing. These definitions are completely reliant on the assumption that the “equals” sign in the Work-Energy Theorem makes both sides of the equation an “if then, then that” statement.

The closest I have come in this case is that energy is simply work yet to be done, but I still have no way of knowing if my intuition is correct or not from the equation. I was able to define force and momentum without relying on prior definitions, but with this, I’ve hit a roadblock. And that roadblock begs the question in the title.

Also, this is my first time using this site, so any tips on using the site to get the most viewers for a question possible would be very appreciated. Thanks in advance for answering the question!

• What physics did you take before AP Physics C? – David White Feb 21 '20 at 19:57

I get that work is essentially a measure of whether a force is “successful” or not in displacing an object in its direction.

"Successful" has no meaning in connection with the definition of work. In its most basic form, work is one of the two basic means of energy transfer between objects, and is the result of the dot product of force times displacement. The other means of energy transfer is heat, which is energy transfer due solely to temperature difference, as you will learn in Thermodynamics.

And, that if said force is successful, the velocity of the object must change in response and gain kinetic energy.

Work on an object does not necessarily result in a change in kinetic energy of an object. It is net work done on an object that results in a change in kinetic energy. The governing principle is the work-energy theorem which states that the net work done on an object results in a change in kinetic energy.

An example where work doesn't necessarily result in a change in kinetic energy is work done against mechanical kinetic friction.

Say you apply a constant force pushing a box at constant velocity on a surface with friction. You do positive work on the box giving it energy since the direction of your force is in the same direction as the displacement of the box. But since the box is moving at constant velocity, the work you do doesn't change the kinetic energy of the box. This is because at the same time the friction force that opposes you is doing an equal amount of negative work since its force is in the opposite direction to the displacement. The net work done on the box is zero and there is no change in kinetic energy. What happened to the energy you gave the box? Friction took it away raising the temperatures of contacting surfaces increasing the internal energy of the box and surface (a.k.a friction heating).

I think that kinetic energy is just the ability to influence another object’s acceleration by motion. But here’s the thing. These definitions are completely reliant on the assumption that the “equals” sign in the Work-Energy Theorem makes both sides of the equation an “if then, then that” statement.

I'm afraid I don't follow you here. But insofar as the "influence" of kinetic energy is concerned, work can certainly result in the transfer of kinetic energy from one object to another. As far as a "sign in the work energy theorem" is concerned, net work can result in either a positive or negative change in kinetic energy. If you bring an object to a stop over a distance "d", the force you apply is opposite the direction of the displacement of the object you bring to a stop. From the work energy theorem

$$W_{net}=F_{ave}d=\frac{mv_{f}^2}{2}-\frac{mv_{i}^2}{2}$$

where $$f$$ and $$i$$ indicate the final and initial velocity of the object, $$d$$ is the stopping distance, and $$F_{ave}$$ is the average force exerted over the stopping distance. Since the final velocity is zero, the work done brining the object to a stop is negative. Negative work means the work has taken energy away from the object.

The closest I have come in this case is that energy is simply work yet to be done, but I still have no way of knowing if my intuition is correct or not from the equation. I was able to define force and momentum without relying on prior definitions, but with this, I’ve hit a roadblock. And that roadblock begs the question in the title.

Perhaps my responses to your previous statements clarifies this. But with regard to the title of your post, the definitions of fundamental terms don't come from the work energy theorem. It's the reverse. The work-energy theorem comes from the definition of work and Newton's second law. Not the other way around.

Hope this helps.

I taught AP Physics C for 10 years, and I have a few suggestions.

1) Take the definition of work at "face value", and don't add anything to it. Considering work to be a measure of how "successful" a force is in displacing an object is very likely to lead to misconceptions and hidden assumptions. By definition, work equals a force multiplied by the distance over which that force acted, and nothing more. In addition, only the component of that force that is in the direction of the displacement adds to work, meaning that carrying an object across a room at constant velocity and constant height does not involve any work, because the force of gravity is acting vertically and you are displacing the object horizontally.

2) An object's velocity does not necessarily change if work is done on it. In the presence of friction, disipative forces are involved, and it is possible for work to be done on an object that moves at constant velocity because friction is turning that work into heat instead of kinetic energy of the object. In that circumstance, the magnitude of the friction force equals the magnitude of the force on the object (or it wouldn't move at constant velocity), the NET force on the object is zero as a result, and the NET work on the object, which is the net force on the object multiplied by the displacement, is zero as a result.

3) Kinetic energy is the energy of motion, as defined by the equation $$KE=\frac{1}{2}mv^2$$, and nothing more. As far as kinetic energy being the ability to influence another object's acceleration by motion, you are very close to forming a concept that ties energy and acceleration together. Force is tied to acceleration via Newton's 2nd law, and energy can be thought of as the ability to do work, but you should not try to form a "mental link" between acceleration and energy.

4) The equals sign is a mathematical idea, which states that there is equivalence between what is on the left hand side of the equal sign and what is on the right hand side of the equal sign. It is not an "if then, then that" statement, which is a computer programming idea.

Regarding physics intuition, the only way to get that is to work a variety of problems. The more problems you work, the more intuitive the physics will become. Also, listen to your teacher regarding the AP released questions and the rubric that was used to grade those questions. The AP graders are looking for very specific things, and the more released questions and rubrics that you review, the better you will be able to identify what those things are when you take the AP test.

I take your question as a question about the nature of kinetic energy.

Let me recount some history:

In the development of mechanics, the study of motion of objects, there was a long period of strong contention as to what should be regarded as the true Quantity of Motion. This conflict arose because there are two candidates:

In modern notation:

$$mv$$

$$\frac{1}{2}mv^2$$

The first is of course the quantity 'momentum' and the second 'kinetic energy'.

Both these quantities comply with the principle of relativity of inertial motion.

The concept of kinetic energy can be visualized as follows: let's say you have a rolling ball, knocking over pins. The pins are arranged in a row, the ball is knocking them down one by one. Every time the ball knocks down a pin it loses some of its velocity. (Assume each pin receives the same impulse.) Given a particular starting velocity the ball will come to a stop after a certain number of pins has been knocked over.

Now compare two runs, and in the second run the ball is given twice as much velocity as in the first run. Then in the second run the ball will knock over 4 times as many pins as compared to the first run. This is of course due to the fact that when accelerating/decelerating distance covered is a quadratic function of time ($$s = \frac{1}{2}at^2$$)

I recommend thinking of kinetic energy in terms of the basic relation ($$s = \frac{1}{2}at^2$$) rather than in terms of some higher level concept such as 'ability to influence another object's acceleration'.

(finishing the history:
Over time the debate among scientist as to the true expression of Quantity of Motion evaporated. Today we use both, and they are regarded as equally important.)

Let me simplify to the case where the force is constant.

The effect of force is of course cumulative. If the duration of the force acting is twice as long it causes twice the amount of change of momentum. That is, change of momentum can be computed by multiplying force with time

Multiplying force with time feels very natural; force causes acceleration, the longer the duration of the acceleration the more velocity.

We regard kinetic energy as a good measure of quantity of motion, so we need an expression to relate force to kinetic energy.

The general definition of work done is an integral, but when the force is constant the calculation simplifies to multiplying the force with the distance traveled.

$$F\Delta s = \Delta(\frac{1}{2}mv^2)$$

For sure: multiplying force with distance doesn't have the natural feel of multiplying force with time.

It is what it is: work done is how you express the relation between force being exerted and kinetic energy. That is: the concept of kinetic energy is what gives the concept of work done meaning.