Kinetic Energy Formula — KE = ½mv²

The kinetic energy formula KE = ½mv² explained: what each variable means, why speed is squared, assumptions and a worked example.

What it means

Kinetic energy grows with the square of speed: doubling the speed quadruples the energy. That is why a modest increase in driving speed makes crashes dramatically more violent — the energy the crumple zones and brakes must absorb scales as $v^2$. The one-half is not a fudge factor; it falls out of the calculus when you add up the work done over the whole acceleration. Mass enters only linearly, so at everyday speeds, going faster matters far more than being heavier.

Variables

Assumptions

• Object treated as a point mass (no rotation considered)
• Speeds far below the speed of light (non-relativistic)
• v is the speed relative to the chosen reference frame

Derivation

Start from the work-energy theorem: the work done accelerating a mass $m$ from rest to speed $v$ is $W = \int F\,dx = \int ma\,dx$. Writing $a = v\,dv/dx$ turns the integral into $m\int v\,dv$, which evaluates to $\tfrac{1}{2}mv^2$.

Worked example

Given:

• m = 70 kg
• v = 8 m/s

1. Write the formula: $KE = \tfrac{1}{2}mv^2$

2. Substitute: $KE = 0.5 \times 70 \times 8^2$

3. Calculate: $8^2 = 64$, so $KE = 35 \times 64 = 2240\ \mathrm{J}$