Kinetic Energy

How do work and energy relate to each other? Consider an object with mass m being acted on by a force F. Then, if the speed of the object increases from v₁ to v₂ as it moves from x₁ to x₂, we can write

solving for a, get

so the force is

Using (40), we find that the resulting work is

(44)

The quantity ½mv² on the right hand side of (44) is called the kinetic energy of the object. Kinetic energy is the energy an object has due to its motion. It is one of three types of energy that can associated with an object. The other two are the potential energy, which is the energy an object has due to its position inside a force field, and the rest energy, which is the energy an object has due to its existence. Returning to (44), we that it tells us that

W = K₂ - K₁ = DK

(45)

i.e., the work done by an external force on an object is equal to the change in the kinetic energy of the object. From (45), we see that kinetic energy (and thus, by conservation of energy, all energy) has units of Joules.

Example:
Consider the box being pulled across the horizontal surface again. If the box has a mass of 10 kg, the coefficient of friction is 0.25, the distance that the box is pulled is 5 m, the force is 150 N and the angle is 30 degrees, what was the initial speed of the box if the final speed is 12 m/sec?

The final kinetic energy is

The work done was found to be

So the initial kinetic energy is

and thus, using the definition of kinetic energy

What if the object is rotating? Since work and kinetic energy are both scalars, we can see that the total work and total kinetic energy is just the sum of the linear work and linear kinetic energy with the rotational work and rotational kinetic energy, respectively, i.e.

and