Einstein and Newton Stop on the Runway
Dave Jesse on January 31, 2013

In my last blog I mentioned two different ways of looking at the ability of an aircraft to stop on the runway, and here is a supplement about dimensional analysis of these measures.

In dimensional analysis we ignore units of measurement, and just express the fundamental parameters being used. So, a speed could be measured in units of knots, miles per hour or metres per second, but dimensionally all speeds are L.T-1 ; they are all length divided by time.

Energy, whether it be the potential energy of an aircraft at altitude, or the kinetic energy of movement will always carry the same dimensions. I always remember the dimensions using Einstein’s famous equation E=m.c2

Therefore the dimensions of energy are M.L2.T-2

One type of runway overrun precursor takes the kinetic energy of the aircraft and divides that by the distance remaining, giving a parameter with the dimensions:
(M.L2.T-2).L-1 = M.L.T-2

Now, L.T-2 is the dimension of acceleration and if we remember what Newton (he of apple fame) told us, force is mass times acceleration, so anything with dimensions of M.L.T-2 is a force.

This will tell us the force required to stop on the runway, but bigger aircraft with bigger masses will necessarily have bigger forces, so it will not be a good measure for comparing stopping distances for large and small aircraft.

Dividing this parameter by the aircraft mass will give a measure with the units of acceleration, e.g. m.s-2 or knots per second. The numerical value with change depending upon the units we select. A cunning extra step is to divide by another parameter with units of acceleration and the obvious one is the acceleration due to gravity. We then have the dimensional formula:
L.T-2 / L.T-2 = 1
or to put it another way, the result has no dimensions and will be the same numerical value irrespective of whether we use imperial, metric or any other system of measurement. Let’s call it k.

What good is a number with no units? Well, it happens that another relevant parameter is the friction on the runway which is expressed as the ratio between the perpendicular force on the runway and the maximum tangential force available. This ratio of forces is largely independent of the size of the forces, and is what a runway friction or µ-meter reads.

Or, if |k| > µ and you try to stop on the runway using brakes, your future is already known. You will be leaving the runway soon…