Kinetic Energy - Physics for Pilot Assessments

Anything moving carries energy because of its movement, and that energy is called kinetic energy. By the end of this lesson you should be able to calculate the kinetic energy of a moving mass, run the calculation backwards to find a speed or a mass, and — the part almost everybody gets wrong — say what happens to the kinetic energy when the speed doubles.

Where the energy of movement comes from

Start from the previous lesson. Push a stationary trolley with a steady force and it accelerates, so while your push lasts the trolley is moving and covering distance; force multiplied by distance means you are doing work on it, and that work has to have gone somewhere. Nothing about the trolley has changed except that it is now moving, so the energy you transferred is stored in the fact of its motion. That is kinetic energy: the energy an object has by virtue of moving, equal to the work that had to be done to get it from rest up to that speed. Stop it again — by braking, or by hitting something — and that same energy comes back out, which is why a moving mass can do damage while a stationary one cannot.

Two things ought to affect how much there is, and both do. A heavier object at the same speed took more force to accelerate and therefore more work, so kinetic energy grows with mass. A faster object was accelerated for longer and through more distance, so kinetic energy grows with speed. What is not obvious is how it grows with speed, and it does not grow in proportion. Here is where it comes from, using the constant-acceleration relationship from the mechanics module, v squared equals u squared plus twice the acceleration times the distance. Starting from rest, u is zero, so v squared is 2ad, and therefore ad is v squared divided by two. Now the work done is force times distance, and force is mass times acceleration, so the work is m × a × d — which is m multiplied by that same ad, and ad has just been shown to be v squared over two. The work done, and so the kinetic energy gained, is half the mass times the speed squared.

Kinetic energy

KE = \tfrac{1}{2} m v^2

$KE$: kinetic energy — the energy of movement, in joules (J)
$m$: mass, in kilograms (kg)
$v$: speed, in metres per second (m/s)

Order of operations matters here more than in any other formula in the module, so it is worth spelling out. Squaring a number means multiplying it by itself, so v squared for a speed of 20 metres per second is 20 × 20 = 400. You square the speed first, then multiply by the mass, then halve the result — and the half applies to everything that came before it, not to the speed alone. Halving the speed and then squaring gives a completely different answer from squaring and then halving, and getting those two steps the wrong way round is the commonest arithmetic error in the whole topic.

Why the square changes everything

Because the speed is squared, kinetic energy climbs far faster than speed does. Double the speed and you multiply the energy by 2 × 2, which is four. Treble it and you multiply by 3 × 3, which is nine. Ten times the speed is a hundred times the energy. Nothing about the object changed in any of those cases — the mass is the same and the half is the same — so the whole increase comes from the squared speed, and the ratio is exact rather than approximate.

Kinetic energy at three speeds for the same mass. The bars stand in the ratio 1 to 4 to 9, because it is the speed that is squared, not doubled or trebled.

You have met this consequence already without the energy vocabulary. In the mechanics module, braking distance was shown to grow with the square of the starting speed, because the relationship linking speed, acceleration and distance contains v squared. It is the same fact wearing a different name: to stop, all of a vehicle's kinetic energy has to be removed by the brakes, the brakes take it away at a roughly steady rate per metre travelled, and four times the energy therefore needs about four times the distance. A car at 40 metres per second carries four times the kinetic energy it has at 20, not twice, which is why the stopping distances in a highway code grow so alarmingly down the page.

Example 1 — the same mass at two speeds

A 2-kilogram ball is thrown at 10 metres per second. What is its kinetic energy? And what would it be at 20 metres per second?

At 10 metres per second, square the speed first: $v^2 = 10 \times 10 = 100$
Now multiply by the mass: $100 \times 2 = 200$ Then halve it: $KE = \frac{200}{2} = 100\ \text{J}$
At 20 metres per second, square again: $v^2 = 20 \times 20 = 400$
Multiply by the mass: $400 \times 2 = 800$ Halve it: $KE = \frac{800}{2} = 400\ \text{J}$
The speed doubled and the energy went from 100 to 400 joules — four times, not twice. If you got 200, you doubled the energy instead of squaring the speed, which is the error this lesson exists to prevent.

Why it matters in aviation

An aircraft moving along a runway is carrying kinetic energy, and because the speed is squared, a small excess of speed is a large excess of energy. Stopping it — at the end of a landing roll, or in a take-off abandoned before the aircraft flies — means removing all of that energy, and most of it leaves through the brakes as heat. Since the brakes take energy away at a roughly steady rate per metre, four times the energy needs about four times the distance, so touching down modestly fast lengthens the stopping distance out of all proportion to the extra few knots.

The same square law is behind the limits published for the brakes themselves. There is a maximum amount of energy an assembly can absorb in a single stop before it becomes too hot, which is why a rejected take-off at high weight and high speed is treated as a demanding event for wheels, brakes and tyres rather than a routine one, and why a cooling period is required before the aircraft can depart again. Judging an approach speed is, in effect, deciding how many joules the brakes will have to swallow.

What the test asks

One misconception dominates this topic, and the bank uses it repeatedly: that doubling the speed doubles the kinetic energy. It quadruples it, and the distractor is always sitting there as exactly half of the right answer — so if an item gives you a speed change and your answer matches one of the options after a single doubling, check whether you squared. The related version substitutes braking: an item states that a vehicle at twice the speed needs twice the stopping distance, which is wrong for precisely the same reason. Expect also a mass-versus-speed comparison of the kind in Example 4, where the reliable trap is picking the heavier object without calculating anything, and one item where the speed is given in kilometres per hour and has to be converted to metres per second before it is squared, since the formula is defined with metres and seconds.