Newtonian mechanics

Kinetic energy

A mass m moving at speed v has kinetic energy K = ½mv²; net-force work changes its kinetic energy.

A moving vehicle can transfer energy when it hits an obstacle; a heavier or faster vehicle can cause a greater change. The energy associated with motion is kinetic energy.

K=\frac12mv^2,\qquad W_{\rm net}=\Delta K

Definition: Kinetic energy

In classical mechanics, a mass $m$ moving at speed $v$ has kinetic energy $K=\frac12mv^2$ (J). It is a nonnegative scalar and depends on the reference frame. The work–kinetic-energy theorem states that the total work on an object equals $\Delta K=K_f-K_i$ .

Vary mass and speed; see the square-speed dependence and how work speeds the object up or slows it down.

Work changes kinetic energy

Positive net work increases kinetic energy; negative work decreases it. For a pull along the motion, an object starting from rest and receiving work $W$ reaches $v=\sqrt{2W/m}$ . Doubling speed quadruples kinetic energy; doubling mass doubles it at fixed speed.

Example: Calculating kinetic energy

A 0.5 kg ball moves at 4 m/s. Find its kinetic energy.

Solution

$K=\frac12(0.5)(4^2)=4\,J$ .

Because kinetic energy depends on speed squared, braking from 20 m/s requires dissipating four times the energy of braking from 10 m/s for the same mass. The work–kinetic-energy theorem says a net work of 150 J increases kinetic energy by 150 J; when a bicycle slows, the resistive forces do negative work. Kinetic energy depends on reference frame: a roadside observer and a passenger measure different speeds and therefore different kinetic energies.

Example: Energy of a bicycle

A bicycle and rider have a combined mass of 50 kg and move at 6 m/s. Find their kinetic energy.

Solution

K=½mv²=½×50×6²=900 J.

A small fast object can have more kinetic energy than a heavy object moving very slowly because speed is squared. Comparing collision effects therefore requires both mass and speed. Brakes remove kinetic energy by converting it mainly into heat.

Quick check

At fixed mass, what happens to kinetic energy if speed doubles?

If the net work on an object is negative, what happens to its kinetic energy?

References

Young, Freedman (2019). University Physics