Kinetic Energy
Energy is the essence of the universe. In the moments after creation there was - according to creation accounts and according to physics - a great deal of light. Think about all the light in all the hundreds of billions of stars within all the hundreds of billions of galaxies that we see today, but contained in a volume smaller than a pinhead. As if that isn’t hard enough to imagine, that is a drop in the bucket as compared to how bright the light at the first moments of the universe was, since it also contained all the energy associated with the rest masses of everything that populates the universe today. We’ll see how much energy that is in this chapter. Eventually, that energy spread out and everything material and otherwise that we have today emerged from it.
Light is at one level just pure energy in the form of electromagnetic radiation, and we will mathematically derive light in our second semester studies. But light also contains momentum, which is somewhat strange given the fact that it has no mass, but that’s a later discussion. What it means to have momentum is that light can hit you and push you back just like other things do when they hit you. Luckily it’s not a lot of momentum, or else we’d be shoved to the ground as we walk outside on a bright sunny day! But we can use that tiny shove from the momentum of light for useful purposes. For instance, for travel within our solar system toward the outer planets, we can deploy solar sails and let sunlight push a space probe outward away from the sun. This was tested recently, as you can see in this link: https://www.planetary.org/sci-tech/lightsail. As a final note, light may also contain angular momentum like spinning object do, but doesn’t always. That also is a more advanced discussion for a later course.
In the present era of the universe, there are many manifestations of energy. Light is still around, but so is mass, and the motion of that mass, and gravitation, and other force fields. Soon we will see that there is energy associated with all of these.
In this chapter we will restrict our focus to single mass systems - meaning systems that we'll assume move as rigid bodies. Macroscopic objects are never really single masses since all the atoms can wiggle, but these details will be ignored for now. A single mass can only possess two types of energy really - either rest energy associated with its mass:
or kinetic energy due to motion at some speed relative to another object or observer. Kinetic energy is defined by:
We will not discuss rest energy right now in much detail, but will postpone that discussion until a course on modern physics. One thing worth saying about it is that there is a tremendous amount of energy associated with even little masses. In this sense you can think of mass as some sort of condensed energy. For instance, it can easily be calculated using the rest energy equation above that a standard aspirin tablet (325mg) has a rest energy equivalent to the energy of combustion of more than 240,000 gallons of gasoline! (A gallon of gasoline releases of energy when it burns.) At 30 miles per gallon in a car, that little aspirin tablet’s rest energy would take you 7.2 million miles! It is rest energy conversion that makes nuclear weapons so devastating. By breaking apart nuclei, mass is converted to a combination of kinetic energy and radiation… and it takes very little mass to release formidable amounts of energy.
Separate from rest energy is kinetic energy due to motion. It is worth noting that kinetic energy is not a property of an object apart from an observer. After all, if you and I are measuring the kinetic energy of a moving ball and if I am moving with respect to you, we won't agree on the ball's speed. If we don't agree on the speed then we won't agree on the kinetic energy. This means kinetic energy is a relative quantity - one that is only meaningfully measured relative to some observer or frame of reference. Usually we will assume the earth's reference frame to be the frame of the observer, unless otherwise specified.
EXAMPLE: Consider the kinetic energy of a car traveling at down a roadway, and how it changes if the car were to double its speed. In the first case . If the speed doubles, we get
Doubling the mass makes the kinetic energy grow only two-fold, but doubling the speed makes it grow four-fold. That's obvious from the equation, but I just wanted to be sure that you didn't miss that point.
One other thing that's worth mentioning is that this present form of our kinetic energy expression is only an approximation of the truth. It is a very good approximation, but one that breaks down as speeds approach the speed of light. The real equation for kinetic energy is more complicated, and asymptotically approaches as where (speed of light in vacuum). As a loose rule, we should be wary of describing kinetic energy with the equation from this chapter if the speed of an object v>c/10, or 10% the speed of light in vacuum. We will discuss the details of how to quantify kinetic energy at speeds higher than this during studies of modern physics.
On Notation (Please read this)
In this chapter we discuss kinetic energy, which we denote K. If you had a prior physics class in high school, there's a good chance you labeled it KE. If that seems ok to you, I want to change your mind. The obvious reason it's such a bad idea is that every time we put two symbols side by side in math, it implies multiplication. When we write F=ma (I know I left off the vector symbols), is there any tendency to think of ma on the right side of the equation as a single variable? Of course not! Why would we think it's ok to put two letters side by side to denote a single quantity? It shouldn't be done. If you don't like K, then another nice way to denote variables of all kinds is with subscripts. is a fine way to denote kinetic energy since we don't read subscripts as products. Maybe even if you want to be really verbose. That way you could describe rest energy as , etc. But please don't ever write KE in homework you turn in from now on.