Kinetic Energy

A bowling ball rolling at 5 m/s and a tennis ball rolling at 5 m/s have wildly different abilities to knock things over. A car at 60 mph isn’t twice as dangerous as the same car at 30 mph — it’s four times as dangerous. A pitched 90 mph fastball carries enough energy to break a bone; a 30 mph toss doesn’t.

What captures these differences is kinetic energy — the energy something has because it’s moving. The amount depends on mass and speed, but speed matters a lot more than you might guess.

The Kinetic Energy Equation

Two things to internalize:

1. KE depends on $v^2$, not $v$. Doubling the speed quadruples KE. Tripling speed makes KE 9× bigger. This is the single most important fact in this section.
2. KE depends on mass linearly. A 2,000 kg truck at 10 m/s has 2× the KE of a 1,000 kg car at 10 m/s. But the same truck at 20 m/s has 4× the KE of itself at 10 m/s. Speed wins by a wide margin.

This is why highway accidents are catastrophically worse than parking-lot fender-benders. A 40 mph crash isn’t twice as bad as a 20 mph crash — it’s four times as bad. Energy scales with the square of speed.

The Work-Energy Theorem

There’s a direct, beautiful connection between work and kinetic energy: the net work done on an object equals its change in kinetic energy.

Applying the Work-Energy Theorem

The theorem is powerful because it bypasses acceleration and time entirely. If you know the forces and the distance, you can find the final speed without ever computing $a$ or $t$.

Worked example. A 1500 kg car traveling at 20 m/s brakes to a stop over 50 m. What is the average braking force?

• $KE_i = \tfrac{1}{2}(1500)(20)^2 = 300{,}000$ J.
• $KE_f = 0$ (the car stops).
• $W_{net} = KE_f - KE_i = -300{,}000$ J.
• Net work also equals $F \cdot d \cdot \cos 180° = -F \cdot 50$ (force opposes motion).
• So $-F \cdot 50 = -300{,}000$ → $F = 6000$ N.

That’s it — no kinematic equations, no need for time. We jumped straight from energy in to force out.

When Net Work is Zero

If the net work on an object is zero, its speed doesn’t change. This doesn’t mean no forces are acting — it means the positive and negative work cancel exactly.

A car cruising at constant 70 mph on a flat highway is the textbook case. The engine pushes forward (positive work). Air resistance and rolling friction push backward (negative work). They cancel. Net work = 0, KE constant, speed steady. Lots of forces, no change in motion.

Similarly, an object in a circular orbit has gravity acting on it constantly — but gravity does zero work because it’s always perpendicular to motion. KE is constant; speed is constant; orbit radius is constant.

Kinetic Energy is a Scalar

Unlike momentum (which is a vector), kinetic energy has no direction. Two cars moving in opposite directions at the same speed have the same KE — even though their momenta cancel out. This is why head-on collisions can completely stop both cars (momenta cancel) while still releasing the combined KE of both into deformation, heat, and sound.

This scalar nature also makes energy easier to work with than momentum in many problems. You don’t need to break it into x and y components.