There are some sets of symbols that, when interpreted, capture something significant about the way things work.
This is one such set:
KE = \frac{1}{2}m\cdot
v^2
Where m is a mass, v its speed and KE its kinetic energy.
This is an equation that receives relatively little attention compared to more famous cousins, yet here we find both a practical tool — a universal exchange rate between energy and speed — and a conceptual statement about the meaning of energy.
The cost of speed
This equation tells us about kinetic energy, the energy associated with the movement of a rigid object. It states that this depends on two things: the mass of that object and the speed of motion, take the first and multiply by the square of the latter, halve that, and done.
It expresses how many units of energy you need to purchase a unit of speed — most interestingly, it specifies that the cost of speed increases with the amount you wish to buy. For purchases of speed, there is an economy of parsimony, and not of scale.
Let’s think about what this means for flying.
A passenger plane accelerates from standstill (no kinetic energy) to around 800km/h. The mass of the plane and contents must be accelerated to this cruising velocity. Ignoring the details of aviation design and the tedious calculations involved in arriving at a more precise relationship, the energy consumed by a plane is proportional to the mass. This is why you are charged for overweight baggage.
What about speed? Do faster planes use less fuel because they arrive sooner at their destination and so spend less time with their engines on? A very simple application of our model that assumes that most energy is consumed in accelerating the plane and very little in maintaining it in the air, would suggest that the full energy use is proportional to mv2 with some small term that depends on flight time as a correction.
This does not feel right — we do not hear the engines cut out during flights. In fact, something is being accelerated all the time: air molecules. The plane is continually pushing against them, and a good thing too, since the air molecules are also pushing back, and due to the cunning design of the wings, this exchange counteracts the downward pull of gravity.
Assuming there is no wind, the air molecules will be moving in all directions but with an average velocity of zero. When the plane hits them their average velocity is going to change to around v.
Let’s pull some threads together so we can find how the total energy used by the plane relates to some parameters like its average speed and the distance travelled. We start with:
TotalFuelConsumption = ConsumptionRate * TimeInAir
But the time in air can be found as the ratio of distance and speed (speed=distance/time) so using d for distance:
FuelConsumption = \frac{ConsumptionRate * d}{v}
Now recall that is the constant acceleration of air molecules that determines the consumption rate. For a small interval of time, we have the consumption being equal to the mass of air displaced in that time, multiplied by velocity squared.
ConsumptionDuringInterval = c\cdot massAirDisplacedDuringInterval \cdot v^2
Where the constant, c, expresses some complicated engineering stuff (and probably isn’t exactly constant). But how much mass of air is displaced per unit of time? It must be proportional to the volume of air the plane travels through in that time — but that is also proportional to the speed of the plane!
ConsumptionRate \propto airDensity⋅v^3
Where we’ve got rid of constants and switched to using the fishlike proportional to symbol. We’ve also used air density rather than mass — which is just mass per unit volume.
FuelConsumptionCruising \propto airDensity \cdot v^2 \cdot d
This is a disaster! It shows us just how horrifyingly inefficient transatlantic jets are:
- Not only do they take you on very long journeys (d is large)
- But they are very fast (v is large, and v2 is even larger)
- We can only console ourselves that air density will be lower higher up
Whether we consider the initial acceleration or the time spent cruising, we get the same relationship between energy and speed. Speed is not just expensive, it is expensive-squared. Would you buy a larger house if the price per square foot increased with size, so doubling your space quadrupled your mortgage?
Undeniably, the true relationships are more complex in both cases (fuel dependence on mass and speed) — for example increasing the mass of airliners by 1% typically results in about 0.75% more fuel consumption, not the 1% expected. Equally vehicles are often less efficient at very low speeds than slightly higher ones — since there’s energy being used in other ways (to keep the motor turning over, or the A/C on). Finally, trying to cruise a passenger jet at low speed simply will not work.
Details are important, but do not let them eclipse a principle that you can apply to any kind of transport — from understanding the range of Teslas (drive slower to get further) to the relationship of braking distance to speed (a parabolic increase, since all the kinetic energy must be transferred to the brakes to decelerate). This is an equation that allows us to cut through many details and arrive at a simple rule:
The key to efficiency is: light and slow.
The meaning of energy
From these practical considerations, we now turn to something rather different. What does this equation tell us about what energy is?
Your first thought might be: it tells me a good deal about kinetic energy, but that’s just one form of the stuff. Absolutely right, but it is still able to tell us something very general that applies to all kinds of energy. It tells us the dimensions of energy.
To understand this, think of the units we use to measure energy.
The (SI) unit of energy is the Joule, but that factoid doesn’t contain much information, it can be rephrased as “the unit of energy is the unit of energy”. We can use our equation to analyse the flavour of a Joule:
KE = \frac{1}{2}m\cdot
v^2
Not only must the left value of the equation equal the right value, the units must be the same.
The crucial information here is that kinetic energy has “dimension” of mass * speed–squared. And if kinetic energy has this dimension then all forms of energy must have it.
This instantly gives you clues about what energy in other scenarios. For example, the energy in a gas will also have this dimension of mass multiplied by speed-squared. What properties of the gas could produce that value? Perhaps the mass of all the molecules multiplied by their average speed-squared.
Things get more puzzling when contemplating the energy of light — photons have no mass, yet they have energy. How can something massless produce something with a mass dimension? The resolution to this conundrum is in a yet more famous relation, one of the famous cousins mentioned at the start:
E = m\cdot c^2
This equation is remarkably close to our hero KE = … instead of a general speed v, we have a constant speed c. It is the interpretation of this, however, that is strikingly different: the above is read as showing a mass-energy equivalence. Indeed it provides a very simple exchange rate which tells us how much energy of some mass will buy you. So the massless photon can have energy — a property that has a mass-like dimension to it — because energy itself is a kind of mass.
It is curious to note that there is nothing in the symbols of our original, Newtonian, equation to stop you from thinking of it as encoding a mass-energy equivalence. The two equations are, we could say, syntactically equivalent but with very different meanings — a type of physical Winograd schema.
Here we have taken a practical tool — what physicists call dimensional analysis — and used (or perhaps abused) it to philosophical ends. This great equation tells us about energy in the form where it is perhaps most comprehensible, and thus offers a foundation layer on which we can build our understanding.
I will not pretend that this way of looking at things solves the mysteries of energy, but I hope it brings them into sharper relief.
Notes
If you enjoy this sort of calculation relating basic physics to energy use, I thoroughly recommend the late David Mackay’s Sustainable Energy Without The Hot Air which has a similar treatment of fuel consumption in planes.
The unimportance of direction
Note that I’ve using the term speed and not velocity above — though normally v stands for the latter. Velocity is a vector quantity, defining the direction of motion as well as the speed. In the case of this equation, the distinction is unimportant, since when you take the square of a vector (its dot product in more technical terms) you lose the vector quality. When squaring, you end up with a scalar who’s value has no dependence on the direction of the velocity vector. In itself this is important, it tells us that kinetic energy (and energy in energy) is directionless, unlike momentum.