Domino Chain Reaction
This activity belongs to the GeoGebra book The Domain of the Time.
This animation simulates, thanks to the domino effect, a real-time chain reaction of increasingly larger tiles or slabs. Now, each piece is larger than the previous one. Even so, the accumulated potential energy allows the domino effect to continue, as shown in this video. The smallest piece is 5 mm tall. Since this height is very low for an acceleration of nearly 10 m/s², it will fall very quickly, while the taller pieces will take longer to fall.
When the presenter states that each piece is approximately one and a half times larger than the previous one, what they actually mean is that each dimension of the piece (thickness, width, and height) is 1.5 times greater than the previous one.
And the smallest domino is about 5 mm high and 1 mm thick (...) If I had 29 dominoes, last domino would be as tall as the Empire State Building.
Indeed, if the height of the smallest piece is around 5 mm, with 29 pieces, the height of the last one would be 5 · 1.528 mm, which is about 426 meters (the height of the Empire State Building is 381 meters). However, this doesn’t mean that 29 pieces are needed to topple the Empire State in this way. Only 26 pieces would suffice. The presenter, referring to the last slab that falls, mentions:It weighs about 100 pounds and is more than a meter tall.
Approximately 100 pounds is about 45 kg. If we then add 13 more pieces in the same progression, making a total of 26 pieces, the last one would weigh 45 · (1.53)13 kg, which is around 331,000 tons, roughly the weight of the Empire State. Moreover, in the video, the height of the last slab should be only 65 cm, since the first piece is only 5 mm tall. Thus, the height of the last piece should be 5 · 1.512 = 649 mm, rather than over a meter. If the pieces strictly followed the 1.5 ratio progression, for the last piece to reach more than a meter in height, the height of the first piece would need to be around 8 mm. Finally, let's examine why the presenter estimates a two billion-fold (two thousand million) increase in the energy from the small push needed to knock down the first piece. Each piece has a potential energy given by , where m is the mass, g is the gravitational constant, and h is the height. Since the heights increase by a factor of 1.5 and the masses by a factor of 1.53, the accumulated potential energy in the 13 pieces is: Around 350 million times the potential energy of the first piece. Now, if we estimate that to knock down the first piece requires only about one-sixth of the energy needed to stand it up, we get an amplification of about 2.1 billion times the energy of the initial push. For better performance, it is recommended to download the GGB file.SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M and control the endpoint. The logical value "trans" records the moment of collision with the next piece.
SetValue(M, M + dt v)
SetValue(trans, If(Mang > Mtrans ∧ N < n, true, false))
SetValue(v, If(trans, (x(vt), 0), vt + dt gt))
SetValue(M, If(trans, Rotate((0, r^(N - 1) h), -Mtrans), M))
SetValue(N, If(trans, N + 1, N))
SetValue(M, If(trans, (0, r^(N - 1) h), M))
StartAnimation(anima, Mang < angMax || N < n)
Author of the activity and GeoGebra construction: Rafael Losada.