Jenna Bratz

Rachel Bauer

Domino RallyHow long does it take for a domino

to fall?

Intro

• First set of dominoes date back to 1120 A.D.

• First used for games of strategy

• Lining them up and knocking them down became increasingly popular in the 1980’s with introduction of the game ‘Domino Rally’

Intro (cont’d)

• Our Goals:– Model the ‘toppling’ time of 1 domino– Model the ‘toppling’ time of 2 dominoes– Extend to 3, 4… or ‘n’ dominoes– Find the optimal distance that minimizes

topple time

Literature

• Charles Bert modeled the topple time of both one and two dominoes using a conservation of energy argument, however the experimental time did not match the predicted time

• Shaw discovered that in a long chain of linearly equally spaced dominoes, only the 5 preceding dominoes contribute to the fall of the current domino

• Math 512 at UD verified Bert’s argument with better data, and also verified Shaw’s argument experimentally

• Heinrich and Lutz modeled molecule cascades, in a similar manner to modeling a chain of falling dominoes.

1 Domino (Experiment)

• Procedure: – Domino was placed on sandpaper (to ensure

no sliding)– A ‘domino toppler’ was used to push the

domino to its balancing point and then to just let it go

– A high speed camera was used to capture the topple time (250 frames per second)

1 Domino (Experiment)

1 Domino (Theory)

1 Domino (Theory)

• Assumptions:– The domino will not ‘slide’ (the pivot point will

remain in the same position throughout the fall)

– The domino will not rotate– The domino will start from an initial velocity of

zero, and will have an initial angle of and a final angle of

1 Domino (Theory)

• Conservation of Energy to obtain a model for theta in terms of time

• Here, kinetic energy is the sum of the rotational kinetic energy, and the translational kinetic energy

• The angle is by the choice of our coordinate system

1 Domino (Theory)

• Because of assumptions, the initial angular velocity is zero.

• Also assume that the final translational kinetic energy is zero since all the energy is transferred into the rotational kinetic energy

• Equation reduces to:

1 Domino (Theory)

• Plug in I, writing w as and taking a second derivative of the equation with respect to time gives the following ODE:

• Can reformulate into a first order system with gives:

1 Domino (Theory)

• Stability Analysis:– Equilibria at – Jacobian:

– After analyzing equilibria, obtain that when n is even, there is an unstable saddle and when n is odd, there is a center

1 Domino (Theory)

• Phase Plane:

1 Domino (Theory)

• Numerical Solution of a particular domino:

• Wanted epsilon to be as small as possible, just enough to knock the domino off balance

• Educated guess of epsilon being 1 degree.

1 Domino (Theory)

1 Domino (Theory)

• Fit a curve to the numerical solution

• Only interested in theta up to Pi/2

1 Domino (Theory)

• Used this fit to estimate the time at exactly Pi/2.

• Theoretical total time for one domino to fall is: .27374 seconds

1 Domino (Analysis)

• Experimental mean time: .2667 seconds

• Theoretical time: .27374 seconds

• Difference is .00704 seconds, only 2.57% error!

2 Dominoes (Experiment)

• Set up 2 equally spaced, equally sized dominoes

• Used ‘domino toppler’

• Did 10 trials, spaced the dominoes at 2.17 cm, which was half the height of the domino

2 Dominoes (Experiment)

2 Dominoes (Experiment)

2 Dominoes (Theory)

• Assumptions– The dominoes will not ‘slide’ (the pivot point

will remain in the same position throughout the fall)

– The dominoes will not rotate– Dominoes are parallel and equally spaced– The second domino will receive a fraction of

the first dominoes horizontal velocity

2 Dominoes (Theory)

• The first domino will hit the second domino at a critical angle,

• Using this critical angle, can find the time at which the first domino hits the second, and from this time,

can obtain the speed at which the first domino is moving

2 Dominoes (Theory)

• Finding the velocity of the first domino at the hitting point, will give the initial velocity of the second domino

• After initial conditions of second domino are obtained, the same model as the one domino case can be used to model the fall time of the second domino

2 Dominoes (Theory)

• Chose a distance of half the domino height of the domino chosen in the first theory, d=.0217 m

• This gives

• We fit a curve to the numerical solution of the angular velocity of the first domino, and found the velocity at this critical angle to be

2 Dominoes (Theory)

• Now, all of this velocity is not going to be transferred to the second domino . In particular, we claim that not all of the horizontal velocity will be transferred

• We discovered that any fraction less than ½ of the first domino’s velocity did not cause the second domino to fall

• So we chose to use exactly half of the horizontal velocity

2 Dominoes (Theory)• Horizontal Velocity is given by:

• We want half of this velocity to be the starting velocity of the second domino. So the new ODE for the second domino becomes:

2 Dominoes (Theory)

• Similar to the one domino case, we found a numerical solution.

• Still want the time it takes the second domino to reach Pi/2, so we fit a curve to the numerical solution for theta.

2 Dominoes (Theory)

• Finding the time at Pi/2 gives t=.40069, and adding this to the time it took the first domino to reach the critical angle

(t=.19805).

• So the total time for two dominoes to fall should be t=.59874 seconds

2 Dominoes (Analysis)

• Theory did not match experiment

• Most likely due to random choice of the amount of horizontal velocity that is transferred

• Adjust the starting velocity of the second domino to match data

2 Dominoes (Full Velocity)

• New assumption: all of the horizontal velocity is transferred to the second domino

• New problem for domino #2

2 Dominoes (Full Velocity)

• Numerical solution is shown below

• Steepness can be seen from the phase plane

2 Dominoes (Full Velocity)

• Found curve of best fit (again) and got new topple time for 2 dominoes to be .35764 seconds

• This number is still higher than experimental time

• Reason could be that the first domino may have had some small initial velocity in the experiment, which would decrease the topple time.

Conclusion

• Model for the topple time of one domino was confirmed by data

• Topple time for 2 dominoes is very dependent on the amount of velocity transferred

• Appears that having all of the horizontal velocity transferred gives an accurate estimate for topple time

Further Work

• Improve 2 domino model

• Model n dominoes– For 3 dominoes, incorporate the effect of both

the first and second dominoes

• Find the distance that minimizes topple time

• Explore different spacings, both nonlinear and not equal spacing

Apologies

• We would like to apologize to Patrick C. Rowe, for not spelling his name in dominoes on top of a slab of jello.