Conservation of energyAQA A-Level Physics: Revision notes

The principle of conservation of energy: energy cannot be created or destroyed; it can only be transferred from one form to another, or from one object to another. The total energy of an isolated system is constant.

In mechanics the key transfers are between gravitational potential energy, kinetic energy and thermal energy (also called internal energy) dissipated by work done against resistive forces. Comparing the energy at the start and at the end is the safest route to every question.

Near the Earth's surface, where $g$ is constant, the change in gravitational potential energy when an object of mass $m$ changes height by $\Delta h$ is

$\Delta E_p=mg\Delta h$

The kinetic energy of an object of mass $m$ moving at speed $v$ is

$E_k=\tfrac12mv^2$

Both are measured in joules. Kinetic energy depends on the speed squared, so doubling the speed quadruples the kinetic energy. For a 0.15 kg ball falling 3.2 m: $\Delta E_p=0.15\times9.81\times3.2=4.7$ J.

If resistive forces are negligible, potential energy lost = kinetic energy gained:

$mg\Delta h=\tfrac12mv^2\;\Rightarrow\;v=\sqrt{2g\Delta h}$

The mass cancels, so the speed after falling a height depends only on the height. A ball dropped from 3.2 m reaches $v=\sqrt{2\times9.81\times3.2}=7.9$ m s⁻¹. A pendulum bob released from 0.20 m above its lowest point has speed $\sqrt{2\times9.81\times0.20}=2.0$ m s⁻¹ at the bottom, whatever its mass.

The same method works up as well as down: an object thrown upwards at speed $v$ rises until its kinetic energy is zero, a height $v^2/2g$.

When resistive forces such as friction and air resistance act, some of the gravitational potential energy is dissipated as thermal energy:

$\text{energy dissipated}=\Delta E_p-\Delta E_k=F_{resistive}\times s$

where $s$ is the distance moved along the path, not the vertical height.

Example: a 70 kg skier descends 150 m vertically along a 600 m slope and reaches 20 m s⁻¹. $\Delta E_p=1.03\times10^5$ J, $\Delta E_k=1.4\times10^4$ J, so $8.9\times10^4$ J is dissipated and the mean resistive force is $8.9\times10^4\div600\approx150$ N.

Bouncing ball. Kinetic energy at the floor is transferred to elastic energy then back to kinetic energy; some is dissipated as thermal energy and sound, so the rebound height is less. A ball dropped from 3.2 m that rebounds to 2.4 m dissipates $0.15\times9.81\times0.8=1.2$ J.

Roller coaster. The carriage cannot rise higher than the release height if resistive forces act; with a resistive force the maximum height reached is less than the starting height, and extra energy is needed to climb a hill higher than the first.

Always describe the sequence: stored potential energy, kinetic energy, then thermal energy in the surroundings, with the total conserved.