Kinetic Energy

In less technical terms, you can think of kinetic energy as an indication of how much it'll hurt if the moving object hits you.

The usual SI unit of kinetic energy, is the unit of mass (kg) multiplied by the unit of velocity (m/sec) squared.

Notice that this is the same as the unit of work, the Joule. All forms of energy can be measured in Joules.

Activities & Practice
to do as you read

1. What is the kinetic energy of a baseball (m=0.145 kg) when thrown at a speed of 25 m/sec?

2. If the speed of a moving object is doubled, what happens to its kinetic energy?

3. If the speed of a moving object is tripled, what happens to its kinetic energy?

4. Use the speed of your own running (found way back when we were first learning about velocity) and your mass to calculate your own kinetic energy as you are running.

Examples

1. What is the KE of a White-tailed Jackrabbit of mass 3.4 kg when running at a speed of 15 m/sec?

(click picture for larger version if you want)

2. How fast does a gerbil (mass = 65 g) have to be moving in order to have the same KE as the jackrabbit above?

Notice that I gave you the gerbil's mass in grams (65 g), but in the calculation you have to put the mass in kilograms (0.065 kg) because the Joule is defined in terms of kilograms. Here's the last step of the example, ignoring the numbers and with just the units...

3. What is the KE of a 2-liter soda bottle when thrown at 1 m/sec. (Because soda is mostly water, and water has a density of 1 kg/L, a 2L bottle will be about 2 kg.)

So, if you want to have a gut feeling for how much energy one Joule is, think "that's the work I do to throw a 2-liter bottle of soda at a speed of 1 meter per second."

White-tailed Jackrabbit. (Wikimedia Commons)

Gerbil. (Wikimedia Commons)

Derivation of the kinetic energy equation. Where did the KE equation come from? Let's see.

Let's say there's an object of mass m, at rest. In order to accelerate it to some speed v, we are going to exert a force on it. Let's say the force we exert is the only force, so is equal to the net force on the object. Now, let's calculate the work we do.

Newton's second law of motion tells us that F_NET can be replaced by m·a. Also the distance, d, is equal to the average velocity times the time interval. If you're starting from rest, the average velocity is half the final velocity, v. This gives us...

Now, recall that the acceleration, a, is equal to Δv/Δt, so let's substitute that in...

The Δt's cancel. Also, because we started from rest and ended with velocity v, then the change in velocity, Δv, is equal to the final velocity v.

And the final simplification yields

This, then, is the work done in accelerating a mass m from rest to speed v. The direction of the velocity doesn't matter; only its magnitude (speed) does. We define the kinetic energy to be equal to this quantity.

Notice also that KE is always positive.

Work-Energy Theorem

Repeating the previous derivation, only with the object having an initial velocity instead of being at rest, leads to the conclusion that

This is called the Work-Energy Theorem. It says that if you add up all the work done by every force acting on an object, that total work equals the change in the object's kinetic energy. If the work is positive (the net force is in the direction of motion) then the object speeds up, gaining kinetic energy. If the work is negative — that happens when the net force is opposite the motion ("backwards") — then the mass slows down, losing KE.

A quick note about thermal energy The atoms and molecules that make up stuff are constantly in motion. In gases (such as air) they move about, bouncing off each other. In solids, they jiggle in place. If you take ½·m·v² of every atom and molecule, and add all that up, you would have the total thermal energy of the object. In practice, what we really deal with is changes in thermal energy, rather than the total amount. The takeaway here is that thermal energy is really kinetic energy in disguise. Each atom has a very small mass, and therefore a very small KE. But objects are made of a VERY large number of atoms and molecules, so the total thermal energy of an object can be a lot.

Additional Activities and Practice

5. The chemical energy in a gallon of gasoline is about 1.3x10⁸ J. If the total energy of the full 12-gallon gas tank of a Scion xB were used to accelerate the 1100-kg car, how fast would it be moving?