Gravitational Potential Energy - Physics for Pilot Assessments

Lift something and you store energy in it, which is released the moment you let go. That stored energy is called gravitational potential energy. By the end of this lesson you should be able to calculate it from a mass and a height, run the calculation backwards for either of them, and say what the height is being measured from — because unlike a mass or a speed, a height is only meaningful relative to something.

Lifting is work done against gravity

From the mechanics module, an object's weight is the force gravity pulls it down with, and it is the mass multiplied by the gravitational field strength: weight = mg, where g is about 10 newtons per kilogram on Earth. Mass and weight are not the same quantity — the mass is the amount of matter, in kilograms, and the weight is a force, in newtons — and this lesson is one of the places where confusing them costs you the answer. To lift something steadily you must pull up with a force equal to its weight; any less and it does not rise, any more and it accelerates upwards. So the force needed is mg, and the distance it is applied through is the height you lift it.

That is just work = force × distance from the first lesson of this module, with the force named. The work you do is the weight multiplied by the height, mg multiplied by h, and by the same argument as for kinetic energy that work has to have gone somewhere. The object is no faster than it was, so it has not become kinetic energy; what has changed is its position, and the energy is stored in the raised object being where it now is. Let go and gravity gives it straight back, converting it into movement all the way down.

Gravitational potential energy

GPE = m g h

$GPE$: gravitational potential energy — the energy stored by height, in joules (J)
$m$: mass, in kilograms (kg)
$g$: gravitational field strength, about 10 newtons per kilogram (N/kg) on Earth
$h$: the height gained, measured vertically, in metres (m)

The g in that expression is what turns a mass into a weight, and leaving it out is the single most common mistake in the topic — it produces an answer in the wrong units that is ten times too small. Two honest notes about it. The true value at the Earth's surface is close to 9.81 newtons per kilogram and varies slightly with latitude and altitude, so 10 is a deliberate rounding that keeps the arithmetic mental; every worked example here says which value it is using. And g is not a property of the object at all: it is a property of the place, which is why the same 5-kilogram mass would store less energy per metre lifted on the Moon, where g is about 1.6 newtons per kilogram.

Because the formula multiplies three numbers, the arithmetic is worth doing in a fixed order to keep it easy: mass times g first, which gives the weight in newtons, then multiply by the height. Taking it that way round means you can always sanity-check the middle number — a 5-kilogram object weighs 50 newtons, and if your intermediate figure is 0.5 or 500 you have slipped a factor of ten somewhere.

Height measured from where?

A mass is 5 kilograms wherever you measure it from, and a speed of 10 metres per second needs no reference point, but a height does. A book on a desk is 0.8 metres above the floor, perhaps 30 metres above the street outside, and some hundreds of metres above sea level, so it cannot be said to have one true amount of gravitational potential energy. What you do instead is choose a reference level, state it, and measure every height in the problem from there — the floor, the ground, the bottom of the swing, the surface of the bench. The choice is free, and provided you keep it consistent throughout a calculation it never affects the answer to a sensible question.

It never affects the answer because what physically matters is always a change in height, not a height. A ball dropped from 5 metres to 1 metre above the floor releases whatever corresponds to 4 metres of fall, and that is true whether you called the floor zero or the ceiling zero. So when a question is about energy being stored or released, look for the vertical distance between the start and the end, and put that into the formula as h. If you happen to choose a reference level above the object, its gravitational potential energy comes out negative, which sounds alarming and means nothing more than being below the line you picked.

Only the vertical distance counts

Push a 20-kilogram box 10 metres up a gentle ramp and it rises 2 metres; carry the same box straight up a 2-metre ladder. In both cases the energy stored by height is the same, because both journeys ended 2 metres above where they started. This follows from the first lesson rather than being a new rule: gravity pulls straight down, so only movement along that line — the vertical part — counts towards work done against it, and the horizontal part of the trip up the ramp contributes nothing to the stored energy. The ramp is easier because the force needed at any moment is smaller, but you apply it over a longer distance, and the stored energy at the top is identical.

Two routes to the same shelf: 10 metres along a gentle ramp, or 2 metres straight up. Gravity only cares about the vertical rise, which is the same 2 metres either way — so the energy stored at the top is identical, 400 joules for a 20-kilogram box. The ramp trades a smaller force for a longer distance; it does not change the height.

Note carefully what the ramp does and does not change. It does not change the gravitational potential energy gained, which depends only on m, g and the vertical rise. It does change the total work you have to do, because a real ramp has friction and some of your effort goes into heat rather than height — which is the subject of the two lessons after this one. For now, when a question asks for the energy stored by lifting, use the vertical height and ignore the path.

Example 5 — between two shelves

A 4-kilogram box is moved from a shelf 0.5 metres above the floor to a shelf 2.5 metres above the floor. How much gravitational potential energy does it gain? Take g as 10 newtons per kilogram.

What matters is the change in height, so subtract the heights first: $h = 2.5 - 0.5 = 2\ \text{m}$
Now use the formula with that difference as h: $GPE\ \text{gained} = 4 \times 10 \times 2 = 80\ \text{J}$
The trap is to use one of the quoted heights on its own — 4 × 10 × 2.5 = 100 joules is the energy relative to the floor, not the energy gained in the move.
Check that the reference level really does not matter: measured from the lower shelf instead, the box goes from 0 metres to 2 metres, and 4 × 10 × 2 is the same 80 joules. Any consistent choice of zero gives the same change.

Why it matters in aviation

Altitude is stored energy, and pilots treat it as a store to be spent. Getting an aircraft to a cruising level puts a large quantity of gravitational potential energy into it, and a descent is the controlled release of that store: with height being given up, the engines can be brought back towards idle while the aircraft keeps its speed, because the energy needed to push through the air is now coming out of the height rather than out of the fuel. That is what makes a descent something to be planned as a profile — so many feet to lose over so many miles — rather than begun when the airfield appears.

The same accounting governs the worst case. If engine thrust is lost, the height already gained is the energy that remains, and it can be traded for distance towards somewhere to land or for speed, but the same metres cannot buy both. It is also why being high or low on a descent profile is not symmetrical: an aircraft above the profile has surplus stored energy and can come down without adding thrust, while one below it cannot recover the missing height for nothing.

What the test asks

The dominant distractor is the missing g: an item gives a mass in kilograms and a height in metres, and one option is simply mass × height, which is the correct answer divided by ten. Ask yourself whether the number you multiplied by the height was a force in newtons; if it was a mass in kilograms, you have not finished. The second trap is the ramp, where the distance travelled along the slope is offered as the height — always take the vertical rise. Expect one item that mentions two floors or two shelf levels and asks for the change in stored energy, where the right move is to subtract the heights first and use the difference, and expect one that quietly gives the height in centimetres. A subtler option sometimes claims that a heavier object stores more energy per metre than a lighter one only because it is harder to lift; that is true but incomplete, and the reason is the formula: the mass appears in it directly.