Conservation of Energy - Physics for Pilot Assessments

Energy is never made and never destroyed — it only moves from one form to another, and the total never changes. That single principle is the most useful thing in this module, because it lets you work out a speed without knowing anything about the forces involved. By the end of this lesson you should be able to name the main forms energy takes, follow it through a conversion from stored to moving and back, calculate a speed from a height and a height from a speed, and explain where energy has gone when it appears to have been lost.

The forms energy takes

You have met two forms already: kinetic energy, carried by anything moving, and gravitational potential energy, stored by anything that has been raised. Potential in that phrase does not mean possible — it means stored, waiting to be released — and there is more than one kind of stored energy. Elastic potential energy is stored in anything stretched, squashed or twisted, from a spring to a bow to a squeezed rubber ball. Chemical energy is stored in the arrangement of atoms in fuel, food and batteries. Beyond the stores there are the forms energy takes while it is on the move or being spread about: thermal energy, which is the energy of the random jiggling of particles inside a warm object and is what we usually mean by heat; sound; light; and electrical energy carried along a wire. Every one of these is measured in joules, because they are all the same quantity in different clothes.

Almost nothing that happens involves only one of them. Strike a match and chemical energy becomes light and thermal energy. Drop a plate and gravitational potential energy becomes kinetic energy, and then, in the instant it lands, sound and thermal energy and the energy needed to break the plate apart. Turn on a torch and chemical energy in the battery becomes electrical energy in the wires and then light — plus, unavoidably, some thermal energy, which is why a torch bulb gets warm. Naming the forms in order is half of answering the qualitative items on this topic.

The principle

Now the law itself, which is worth stating carefully because its wording is what the distractors attack. Energy cannot be created or destroyed; it can only be transferred from one object to another or converted from one form into another. Add up all the energy in a system before something happens, add it all up afterwards, and the two totals are equal. A system here just means whatever you have decided to include in the accounting, and it is called closed when nothing gets in or out — no energy leaking away as heat to the surroundings, nothing being added from outside. For a swinging pendulum or a falling ball, treating the system as closed for the length of the fall is a good enough approximation to give the right answer.

Conservation of energy between two moments

KE_1 + GPE_1 = KE_2 + GPE_2

$KE_1$: kinetic energy at the first moment, in joules (J)
$GPE_1$: gravitational potential energy at the first moment, in joules (J)
$KE_2$: kinetic energy at the second moment, in joules (J)
$GPE_2$: gravitational potential energy at the second moment, in joules (J)

Used in that form the principle becomes a calculating tool: whatever gravitational potential energy is lost between the two moments must appear as kinetic energy, so you can find a speed from a height without ever mentioning a force, an acceleration or a time. That is worth pausing on, because the mechanics module would have needed all three.

The pendulum

A pendulum is the cleanest demonstration there is. Pull the bob to one side and let go: at the moment of release it is as high as it will get and momentarily still, so all of its energy is gravitational potential and none of it is kinetic. It swings down, losing height and gaining speed, and every joule of gravitational potential energy it gives up turns into a joule of kinetic energy. At the lowest point of the swing it is at its fastest and has no height left to give, so the store is empty and everything is kinetic. It then climbs the other side, spending kinetic energy to buy height back, until it is momentarily still again at the same height it started from — all gravitational potential once more, and the cycle repeats.

The total energy is the same at every position; only the split changes. All of it is stored by height at the two ends of the swing, and all of it is carried by movement at the lowest point.

Two details in that description are examined directly. The first is that the bob is genuinely stationary for an instant at each end of the swing — it has to be, because it stops going one way before it starts coming back — which is why the kinetic energy there is zero rather than merely small. The second is that the halfway position is genuinely half and half: drop through half the height and you have released half the stored energy, so half remains as gravitational potential energy and half has become kinetic. Nothing about the swing is lopsided, and nothing about the split depends on the mass of the bob.

Example 2 — from speed back to height

A ball leaves your hand travelling straight up at 20 metres per second. Ignoring air resistance, how high does it go? Take g as 10 newtons per kilogram and note that the mass is not given.

At the top of its flight the ball is momentarily still, so all of its kinetic energy has become gravitational potential energy: ½ m v² = m g h.
Every term on both sides contains the mass, so divide both sides by m. The masses cancel and vanish from the problem entirely, leaving ½ v² = g h — which is why the question did not need to give you a mass.
Substitute the numbers: ½ × 20² = 10 × h, and 20² = 400, so ½ × 400 = 200 = 10 × h.
Divide both sides by 10: $h = \frac{200}{10} = 20\ \text{m}$
Check it forwards: falling 20 metres releases g h = 10 × 20 = 200 joules per kilogram, and ½ × 20² = 200 joules per kilogram of kinetic energy. They agree.

That cancellation in Example 2 is worth keeping. Because both the stored energy and the kinetic energy are proportional to the mass, the mass drops out of any pure conversion between the two, and the speed reached after falling a given height is the same for every object — which is exactly the free-fall result from the mechanics module, arrived at a different way. Setting ½ v squared equal to g h and rearranging for the speed gives a compact form worth having.

Speed gained by falling a height h

v = \sqrt{2 g h}

$v$: the speed reached, in metres per second (m/s)
$g$: gravitational field strength, about 10 newtons per kilogram (N/kg) on Earth
$h$: the height fallen, in metres (m)
$\sqrt{\ }$: the square root — the number which, multiplied by itself, gives what is inside

It contains no mass, which is the point, and it says that four times the drop gives twice the speed rather than four times, because of the square root. Take the 5-metre drop from Example 1: 2 × 10 × 5 = 100, and the square root of 100 is 10 metres per second, matching the long way round exactly.

Energy stored in a spring

The third store in that panel needs its own explanation. From the materials lesson in the matter and fluids module, a spring obeys Hooke's law while it is not overstretched: the force needed is proportional to the extension, F = kx, where x is how far it has been stretched from its natural length and k is the spring constant in newtons per metre — a stiff spring has a large k. Because the force needed grows steadily from nothing at the start to its full value kx at the end, the average force applied over the whole stretch is half the final force. Work is force times distance, so the work stored is that average force, ½kx, multiplied by the distance stretched, x — which gives one half k x squared.

Elastic potential energy in a stretched spring

E_\text{elastic} = \tfrac{1}{2} k x^2

$E_\text{elastic}$: the energy stored by stretching or squashing, in joules (J)
$k$: the spring constant — the force needed per metre of stretch, in newtons per metre (N/m)
$x$: the extension away from the natural length, in metres (m)

Notice that the extension is squared, exactly as the speed is in kinetic energy, so stretching a spring twice as far stores four times the energy rather than twice — the same non-linearity in a different setting. Notice too that this is only valid while Hooke's law holds; stretch the spring past its elastic limit and the neat proportionality is gone, along with the tidy formula. Released, that stored energy goes wherever the spring can push it, which for a catapult or a spring-loaded launcher means straight into kinetic energy.

Example 3 — a spring launching a mass

A spring of spring constant 200 newtons per metre is compressed by 0.1 metres and used to launch a 0.4-kilogram trolley along a smooth level track. How much energy is stored, and how fast does the trolley leave?

Square the compression first: $x^2 = 0.1 \times 0.1 = 0.01$
Multiply by the spring constant: $200 \times 0.01 = 2$ Then halve it: $\frac{2}{2} = 1$ joule stored.
On a smooth level track no height is gained and nothing is lost to friction, so all 1 joule becomes kinetic energy: 1 = ½ × 0.4 × v².
Half of 0.4 is 0.2, so 1 = 0.2 × v². Divide both sides by 0.2: $v^2 = \frac{1}{0.2} = 5$
The speed is the square root of 5, which is a little over 2.2 metres per second — and this is a case where the honest answer is not a whole number. Test items are normally arranged so that it is.

Where the energy seems to go

Real pendulums stop swinging. Real balls do not quite bounce back to the height they fell from. Nothing about that contradicts conservation of energy; it means the accounting was incomplete. As the bob moves it drags air past itself and flexes its pivot, and as a ball deforms on impact its material is worked back and forth — and in every one of those processes the energy that disappears from the tidy kinetic and stored columns reappears as thermal energy in the air, the pivot, the ball and the floor, plus a little sound. The totals still balance to the last joule. They simply balance across a longer list.

This is why the phrase energy is lost is best avoided, and why the correct word is dissipated — spread out. The energy that friction turns into thermal energy has not been destroyed; it has been divided among an enormous number of particles in the surroundings, each of which now jiggles very slightly faster. What has been lost is not the energy but its usefulness: concentrated energy in one place can be made to do work, while the same number of joules smeared thinly through a room and the air above it cannot be gathered back up. Rub your hands together and you can feel exactly where the kinetic energy went; you cannot rub them the other way and get the movement back.

Example 4 — the energy that did not arrive

A 2-kilogram ball is dropped from 5 metres, and measurements show it is travelling at 8 metres per second just before it lands rather than the 10 metres per second of Example 1. Where did the difference go, and how much of it was there? Take g as 10 newtons per kilogram.

The gravitational potential energy released is the same as before: $GPE = 2 \times 10 \times 5 = 100\ \text{J}$
The kinetic energy it actually arrives with is ½ × 2 × 8² = 1 × 64 = 64 joules.
The difference is 100 − 64 = 36 joules.
Those 36 joules were transferred to the air the ball pushed out of its way, ending up as thermal energy in that air and in the ball, plus a small amount of sound. Nothing was destroyed: 64 joules of movement plus 36 joules dissipated is the 100 joules the ball started with.
Note that the missing energy is not the difference in the speeds. The speeds fell by a fifth, from 10 to 8, while the energy fell by more than a third, from 100 to 64 — because the speed is squared.

Why it matters in aviation

Because height and speed are convertible, neither on its own tells a pilot very much; the useful quantity is the total of the two. Pitch moves energy between the two accounts, trading height for speed or speed for height, while thrust and drag are the only things that change the sum. Flying an approach well is largely a matter of watching that sum rather than either part of it, and this is what is meant by energy management.

The consequence is worth stating plainly. An aircraft arriving both high and fast has more total energy than the runway ahead of it can absorb, and no amount of pitching will remove the surplus, because pitching only shifts energy from one form into the other — raise the nose and the excess speed becomes excess height, lower it and the height comes back as speed. The surplus has to be dissipated instead, by reducing thrust and by deliberately adding drag with speedbrakes, gear and flap, all of which pass energy into the airflow as heat. Noticing early is what makes it recoverable, since dissipation needs distance to work in; the same excess found late leaves nowhere to spend it, and a go-around is the only way to reset the account.

What the test asks

The central distractor is the phrase energy is used up, or lost, or runs out. An option describing a swing that gradually stops by having exhausted its energy sounds reasonable and is wrong: the energy has been converted to thermal energy in the air and the pivot, and any option that ends with energy simply ceasing to exist can be eliminated on sight. The mirror image also appears — an option in which a device produces more energy than was put into it, which conservation forbids outright. Expect a pendulum position item, where the two reliable traps are claiming the bob has some kinetic energy left at the ends of the swing and claiming it still has gravitational potential energy at the bottom, and expect a numeric drop item where the trap is comparing speeds rather than energies. One last one: because the mass cancels in a pure fall, an item may give you two different masses dropped from the same height and offer the heavier one as the faster. Ignoring air resistance they arrive together.