Kinetic Energy

What kinetic energy is

Any moving body has the capacity to do work by virtue of its motion. This capacity is called kinetic energy:

$KE = \frac{1}{2}mv^2$

It depends on mass and the square of speed. Doubling the speed quadruples the kinetic energy.

Derivation from work-energy theorem

A body of mass $m$ starts from rest ($v_i = 0$) and a net force accelerates it to speed $v$.

Work done by net force = change in KE:

$W_{net} = KE_f - KE_i = KE_f - 0 = KE_f$

Also, from kinematics ($v^2 = 2as$ for constant $a$, starting from rest):

$W_{net} = F \cdot s = ma \cdot s = m \cdot \frac{v^2}{2s} \cdot s = \frac{1}{2}mv^2$

Therefore:

$\boxed{KE = \frac{1}{2}mv^2}$

Units

$[KE] = \text{kg} \cdot (\text{m/s})^2 = \text{kg} \cdot \text{m}^2/\text{s}^2 = \text{J (Joule)}$

Same units as work — consistent with the work-energy theorem.

Key properties

Always non-negative: $KE = \frac{1}{2}mv^2 \geq 0$. Speed is always positive, so KE is never negative.

Scalar: KE has no direction. A body moving north and one moving south at the same speed have the same KE.

Frame-dependent: KE depends on the reference frame. A ball at rest on a moving train has zero KE in the train's frame but nonzero KE in the ground frame.

Relation to momentum

Since $p = mv$:

$KE = \frac{1}{2}mv^2 = \frac{(mv)^2}{2m} = \frac{p^2}{2m}$

This form $KE = \frac{p^2}{2m}$ is extremely useful in collision problems where momentum is conserved.

Effect of speed on KE

This quadratic dependence is why high-speed collisions are so much more destructive — doubling impact speed quadruples the kinetic energy that must be absorbed.