Potential energy

Potential energy (U) is energy a body possesses purely because of where it is or what state it is in — not because it is moving. It is the opposite partner of kinetic energy, which depends on motion. At this level you meet two kinds of potential energy.

Gravitational potential energy is the energy a body has because of its height above some reference level (almost always taken as the ground), on account of the gravitational pull acting on it. This is the kind with a formula:

where: $U$ is the gravitational potential energy (joules, J), $m$ is the mass of the body (kg), $g$ is the acceleration due to gravity, taken as $g = 9.8\ \text{m/s}^2$ unless a question states otherwise, and $h$ is the height of the body above the chosen reference level (m).

SI unit: since $U = mgh$ combines kilograms, metres per second squared, and metres, the unit works out to $\text{kg}\cdot\text{m}^2/\text{s}^2$, which is defined as the joule (J) — the same unit used for kinetic energy and work, because potential energy is, after all, just another form of mechanical energy. Two things control the size of $U$: mass and height, and they matter equally — doubling either one doubles the potential energy. A heavier object is not automatically "more powerful" than a lighter one; a 1 kg book on a 10 m shelf and a 10 kg book on a 1 m shelf store exactly the same gravitational potential energy, because $mgh$ treats both factors the same way.

Elastic potential energy works on a completely different trigger: not height, but a change in shape. Stretch a rubber band, compress the spring in a toy gun, or bend an archer's bow, and you store energy in the deformed material itself. Release it, and that stored energy converts into motion — the pellet flies, the arrow shoots forward. At this level you are only asked to recognise and describe such situations in words; no formula for elastic potential energy is part of the syllabus. A handy way to keep the two apart: gravitational potential energy asks where (how high), while elastic potential energy asks how (how deformed). Both, though, share the same essential character — energy parked in a body, waiting for a chance to become motion.

Question: A flowerpot of mass 3 kg is placed on a balcony 5 m above the ground. Calculate its gravitational potential energy relative to the ground. Then find how this potential energy changes if the same flowerpot is instead placed on a balcony twice as high. (Take $g = 9.8\ \text{m/s}^2$.)

Step 1. Identify the given values for the first case: $m = 3\ \text{kg}$, $h = 5\ \text{m}$, $g = 9.8\ \text{m/s}^2$, and write down the formula, $U = mgh$.

Step 2. Substitute the values: $U = 3 \times 9.8 \times 5$.

Step 3. Work it out in stages: $3 \times 9.8 = 29.4$, and $29.4 \times 5 = 147$. So $U = 147\ \text{J}$.

Step 4. For the second balcony, the height doubles to $h = 10\ \text{m}$, while $m$ and $g$ stay the same. Since $U = mgh$ is directly proportional to $h$, the potential energy simply doubles too: $U' = 3 \times 9.8 \times 10 = 294\ \text{J}$. The flowerpot has 147 J of gravitational potential energy on the first balcony, and 294 J — exactly double — on a balcony twice as high, confirming that $U$ scales directly with height when mass stays fixed.

Competitive stretch

Two identical 2 kg bricks sit on shelves at 1 m and 3 m. Their potential energies are $U_1 = 2 \times 9.8 \times 1 = 19.6\ \text{J}$ and $U_2 = 2 \times 9.8 \times 3 = 58.8\ \text{J}$. Since $U_2 > U_1$, the brick on the higher shelf would strike the ground with more energy if both fell freely, because its larger stored potential energy converts into kinetic energy as it falls.

• Forgetting that potential energy is always measured relative to a chosen reference level — height must be measured from a stated ground or surface, never from thin air.
• Mixing up gravitational potential energy (has the formula $U = mgh$) with elastic potential energy (a stretched spring or bow), which stays qualitative at this level with no formula to substitute into.
• Leaving out $g$ entirely, or using an inconsistent value — always use $g = 9.8\ \text{m/s}^2$ unless the question specifies otherwise.
• Assuming a heavier object always has more potential energy than a lighter one — mass and height both matter, and a small mass raised very high can store more energy than a large mass raised only a little.

Ask the tutor: Want to practise picking the right reference height in a tricky multi-shelf problem, or see how gravitational potential energy converts into kinetic energy as something falls? Ask the AI tutor to work through a few extra examples with you.