Kinetic energy and variable forces

Kinetic energy is the energy a body has because it is moving: K = ½mv². It equals the work needed to bring the body from rest to speed v, or the work it can do while being stopped. It is a scalar and is never negative.

Since p = mv, kinetic energy can also be written K = p²/2m, and p = √(2mK). So if a body's momentum is doubled, its kinetic energy becomes four times as big.

Speed counts twice in K = ½mv². Double the speed and the energy goes up four times; this is why a car crash at 80 km/h is so much worse than one at 40 km/h. The other way round, a body that keeps only a quarter of its energy has kept half its speed.

Two bodies with the same kinetic energy: the lighter one is faster, v = √(2K/m).

What if the force changes as the body moves? Then split the motion into tiny steps, find the work in each, and add them: W = ∫F dx. On a graph, this is the area under the F–x curve:

• Area above the x-axis counts as positive work.
• Area below the x-axis counts as negative work.
• A rectangle, a triangle or a trapezium can be added piece by piece.

Example: a force F = 7 – 2x + 3x² N moves a body from x = 0 to x = 5 m. W = ∫(7 – 2x + 3x²) dx from 0 to 5 = 7 × 5 – 5² + 5³ = 35 – 25 + 125 = 135 J.

The work–energy theorem holds for a varying force too: K_f – K_i = ∫F dx. So if you know how a body's speed changes, you know the total work done on it, even when you do not know the force.