Why is it that dominoes placed closer together fall over faster than those placed futher apart. In a trial where lines of 50 dominoes were placed 2cm,2.5cm ,3cm ,3.5cm and 4cm apart, the ones spaced 2cm apart fell over the fastest. Why is this? What determines the rate at which they fall?
(Also as a side note, the ones spaced 4 cm apart picked up speed and fell over faster than those place 3.5 cm apart.)
Would love an answer from a physics pro. Thank you.
Imagine the extreme cases. Dominoes placed zero distance apart (unspaced), and dominoes placed further apart than their length (over-spaced).
It can be seen that when they are unspaced, the last domino falls over at the same time as the first domino.
When they are over-spaced, the last domino never falls (nor do any other than the first which is directly pushed). Between the two cases, it can be seen there is a relationship between practically instantaneous transmission of force from the first domino, and maximal delay.
What you are observing then is the variability in what is called the "phase velocity" according to the wavelength. That is, the varying speed with which waves propagate in a medium according to their wavelength. In this case, the "medium" of the wave is made of dominoes.
The unspaced dominoes have a wavelength of zero (instantaneous propagation of force from one end to the other), and the over-spaced dominoes have an infinite wavelength (i.e. the wave does not propagate).
As for why those spaced at 4cm travel faster than those at 3.5cm, I can't quite model that, but it's possible that the first domino acquires extra energy (due to the extra distance it falls under gravity and its own weight), and because in the 4cm case the head of the prior domino strikes the next domino closer to the table top (and therefore closer to its pivot point) than in the 3.5cm case, the next domino falls faster and the wave propagation speed increases.
It's not dissimilar to how putting your foot against an upright garden rake, and using the pressure of your foot against the lower part of the handle to lash it to the ground (pivoting around the rake-head), causes the top of the handle to lash toward the ground far more quickly than if you push the upper handle with your hand. (The same basic effect is involved when you stand on the fingers of the rake and the handle lashes into your face.)
Of course, pushing the rake handle at it's lower point requires a lot more force to accelerate it at the same speed, but the key in the 4cm domino case (versus the 3.5cm case) is that this extra force is acquired from the gravitational gain.
So we have a loss of phase velocity that is increased with separation, but we also have this "gravitational gain" that is increased with separation.
But there will be a different trigonometric relationship between each, because the gravitational gain is negative until the centre of gravity of the domino passes over its pivot point (i.e. gravity will resist the falling of the domino until it passes over the pivot), whereas the phase velocity "loss" is always positive. This is why dominoes will rebound into the upright state if the force applied is not sufficient to pushed them over their pivot, because the gravitational gain is negative.
In terms of gravitational gain, there will also be an equilibrium with inertia and air resistance (which prevents each successive domino in the domino wave accelerating indefinitely, and counters the gravitational gain from each step), which I can't quantify but can state in principle.
What this shows is that there is likely to be a sweet spot in the middle of the available range of spacings, where the wave travels slowest - that is, at close spacings, the short wavelength predominates to increase the phase velocity. At long spacings, the gravitational gain and the "levering" effect of the preceding domino striking the next domino closer to its base predominates to increase the propagation speed. In the middle (or to be more precise, at a separation distance that is determined by the thickness and height of the domino), you have the worst of both worlds - a longer wavelength, but relatively little gravitational gain.
