Collision of Billiard Balls
Snap with your fingers a mare’s nose – she will wag her tail.
Absolutely elastic collision is the collision model where the total kinetic energy of the system is preserved. Since the deformations of the bodies are neglected, it is considered that energy is not lost on deformation, and that the interaction is immediately distributed over the entire body. The example of the collision of billiard balls is often drawn to illustrate the case of absolutely elastic collision. However, is the deformation of the balls during their collision, in fact, that negligible, and is the time of their interaction, in fact, that short?
The time of interaction of billiard balls and their deformation, in addition to direct measurement, can be determined from the formulas describing the collision of arbitrary balls of mass M, radius r, with elastic modulus E and Poisson’s ratio γ.
Thus, the time of interaction , and the maximum convergence of balls , where .
The maximum convergence is a conventional term for the difference between the sum of the radii of the balls and the minimum distance between their centers during the collision. These formulas can be found in the world famous textbook on Physics by Lev Landau and Evgeny Lifshitz (Course of Theoretical Physics, vol. 7: Theory of Elasticity, p. 50).
For calculating τ and h0 we need to know the Poisson’s ratio and the modulus of elasticity (Young’s modulus). For most plastic materials, Poisson’s ratio is between 0.3 and 0.35. In his article, “Modeling the Effects of Velocity, Spin, Frictional Coefficient, and Impact Angle on Deflection Angle in Near-Elastic Collisions of Phenolic Resin Spheres,” (http://arxiv.org/ftp/physics/papers/0402/0402008.pdf), Sam Crown gave a value of Poisson’s ratio for a billiard ball γ = 0.34 and Young’s modulus E = 5.84 GPa, without specifying, however, his method for finding out these values.
Currently, there exist two basic methods of measuring the modulus of elasticity (Young’s modulus). The first way is a direct measurement of sample deformation under compression or extension. The second way is to calculate the modulus of elasticity using the speed of sound propagation in the sample , where c is the speed of sound, and ρ is the density of the material.
For measuring the modulus of elasticity using the speed of sound, we used a billiard ball with a diameter of 60 mm, which on both sides was filed down on a lathe to the size of 50 mm. Two piezoelectric elements and contact groups for them were set on the filed planes. Actually, this simple construction is evident from the assembly drawing and a photograph.
If you lightly hit the flat end of the ball, both of the piezoelectric elements will react with characteristic signals with a small delay in time. The time interval between points A and B is 27–30 µs and corresponds to the time needed for the compression wave to propagate from one end of the ball to the other (L = 50 mm).
The speed of sound, calculated according to these values, amounts to about 1760 m / s. It is easy to calculate that with a density of this particular billiard ball of about 1768 kg/m3, the modulus of elasticity will be 5.48 GPa.
For measuring the modulus of elasticity (Young’s modulus) using the direct method (that is, the immediate strain), a sample with a diameter of 32.82 mm and a length of 50.20 mm was made from a billiard ball on a lathe.
The modulus of elasticity was determined using the INSTRON 3369 Tester, which allows for any measurements associated with compression, bending, and breaking of the sample. The maximum compression load is 5 tons. As it turned out, this load was not enough to destroy the sample.
Here is a chart obtained in the process of compressing the sample in the strain range that did not exceed 0.2 percent.
The modulus of elasticity, determined from two tests was found to be 5.30 Gpa. This is slightly less than the modulus of elasticity of 5.48 GPa, as measured by the speed of sound in the material, and correlates well with the magnitude of 5.84 GPa which Sam Crown mentions in his work.
Having determined the modulus of elasticity for a billiard ball as being E = 5.4 GPa, the Poisson’s ratio γ = 0.34, mass M = 0.282 kg, the radius r = 0.068 m, the speed of the cue ball V = 7 m / s, and using the above formulas, we obtain the interaction time of the balls τ = 320 µs and the maximum convergence of the balls h0 = 0.76 mm (which corresponds to a deformation of 0.38 mm for each of the balls).
For confirming these results, high-speed video of the process of billiard ball collision was carried out. The video was performed using a digital high-speed monochrome camera Phantom V12, at a frequency of 49,000 frames per second. The average contact time of billiard balls with a diameter of 68 mm, obtained from four takes, was about 244–268 µs with an impact velocity of about 7 m/s.
Another way of determining the length of the collision could be to measure the time of the closed state of the contact group, attached to the billiard balls.
However, the inability to control the speed of the balls before the collision may give only an approximate value of contact time, which amounted to about 280–300 µs.
Having determined the position of the balls before and after their collision on the basis of high-speed video, we can find the value of their closest convergence h0 = 0.6–0.7 mm, as well as deformation of each of the balls ∆х = 0.3–0.35 mm.
Given that the calculated time of the interaction between the balls and the magnitude of their maximum convergence agrees well with the results of high-speed video and measurements using the contact group, we can consider the values of the modulus of elasticity E = 5.3–5.5 GPa, the interaction time of the balls τ = 250–300 µs, and the degree of their deformation ∆х = 0.3–0.35 mm quite reliable at the collision velocity V = 7 m/s.
After a simple algebraic transformation, we can write the equation for the time of interaction between the balls and the size of their closest convergence in the following form:
and respectively, where c is the speed of sound in the material.
Analyzing the equations written in this form, we can see that the duration of the collision is mostly determined by the size of the colliding balls and the speed of sound in the material, rather than by the speed of the balls themselves. Thus, a decrease in the speed of convergence of balls by two and a half times will increase the duration of the collision of balls by only 17 percent. At the same time, the degree of deformation of the balls already significantly depends not only on their size and the speed of sound, but also on their speed of collision. It is also easy to see that a transition from balls with a diameter of 68 mm to balls with a diameter of 60 mm will reduce both the time of collision, and the degree of deformation by 12 percent. The proof of this is a high-speed video of the collision between balls with a diameter of 60 mm at a speed of 2.8 m/s.
One of the possible extensions of this study could be the comparison of the characteristics of modern billiard balls from various manufacturers with the properties of ivory balls.