University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln

Robert G. Fuller Publications and Presentations Research Papers in Physics and Astronomy

January 1999

Reality and Theory in a Collision

Norman Derby Bennington College, Bennington, VT

Robert Fuller [email protected]

Follow this and additional works at: https://digitalcommons.unl.edu/physicsfuller

Part of the Physics Commons

Derby, Norman and Fuller, Robert, "Reality and Theory in a Collision" (1999). Robert G. Fuller Publications and Presentations. 42. https://digitalcommons.unl.edu/physicsfuller/42

This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert G. Fuller Publications and Presentations by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Note

Reality and Theory in a Collision Norman Derby, Bennington College, Bennington, VT 05201; [email protected] and Robert Fuller, University of Nebraska – Lincoln, Lincoln, NE 68588-0111; [email protected]

The Film and the Theory B. Simplified Model illiard-ball collisions are fre- quently cited in introductory A simple model frequently adopt- A. Description of the Video Clip Bphysics textbooks, usually as ed in textbooks dealing with such col- examples of elastic collisions. Many The Physics: Cinema Classics lisions is (1) to assume it to be per- articles describing such collisions disc provides a Project Physics film fectly elastic (i.e., kinetic energy is have appeared in this journal and of a head-on collision between two conserved), (2) to ignore any friction elsewhere,1–6 but comparisons billiard balls shot nominally at 3000 during the short collision time, and between theoretical results and actual frames per second by a fixed camera. (3) to assume the surface friction collisions are rare, and most of the Each ball has a wide stripe around a subsequent to the collision to be the theoretical analyses have simply diameter making rotation visible, as same for both balls. assumed friction to be negligible dur- shown in Fig. 1. The film shows a cue The solution of this model is ing the collision time. Students trying ball rolling without skidding across a straightforward. Immediately after to simulate billiard-ball collisions on level surface. It then strikes a station- the collision, the cue ball has lost all ary target ball its linear momentum but its angular head on. What momentum has not changed; mean- happens next while the target ball has acquired all can really star- the linear momentum but has no tle a class of angular momentum yet. As time pass- complacent es, the frictional forces between each physics stu- ball and the surface change the linear dents who have and angular momenta of each ball just finished until its linear and angular velocities studying colli- satisfy the no-skidding condition, ␻ = Fig. 1. Points used to determine position and rotation angle of colliding bil- ␶ liard balls in VideoPoint. sions between v/r. Letting i be the duration of the particles and skidding for each ball and f the sur- have seen dem- face frictional force, we obtain two ␶ onstrations of pairs of equations for i and the final a popular simulation program such as Newton’s cradle. As they watch, the linear velocities vif: Interactive Physics® may encounter target ball starts to skid across the interesting collisions, such as ones in surface without any initial spin, but it Linear Momentum: which the cue ball is thrown up into gradually slows down as its spin Cue Ball Target Ball ␶ ␶ the air when the frictional coeffi- increases. Meanwhile, the cue ball f c = mvcf f T = mv0 – mvTf cients are set at even modest levels. slows down sharply at first while We decided to use a video-analysis continuing to spin, and then it begins Angular Momentum: tool designed for the physics class- to accelerate in pursuit of the target Cue Ball VideoPoint® ball. Friction between the balls and room ( ) to examine a 2 slow-motion film of a head-on colli- the surface is, of course, the gremlin fr␶ = ᎏᎏ(mv r – mv r) c 5 0 cf sion made some years ago and cur- responsible for such behavior. During rently available on Physics: Cinema the collision, the frictional torque on Target Ball Classics 7 each ball was relatively small, so . In this paper, we compare 2 some of the experimental data we there was little change in a ball’s rota- fr␶ = ᎏᎏ (mv r) T 5 Tf obtained with theoretical results and tion during the extremely brief colli- the simulation results. sion time, but eventually the effect of These are readily solved yielding such torques became substantial.

24 THE PHYSICS TEACHER Vol. 37, Jan. 1999 Reality and Theory in a Collision ᎏ2ᎏ ᎏ5ᎏ vcf = vo and vTf = vo use the computer mouse to mark the 7 7 position of interesting points. The f␶ 2 f␶ ᎏᎏc ᎏᎏ ᎏᎏT coordinates of the points are then and = vo = ; that m 7 m stored in tables. This makes it easy to is, the skidding times are, take data from many frames of the interestingly enough, the film and to save it in a format that is same for both balls. convenient for analysis. We collected data much as stu- Data dents might by marking three points on each ball as shown in Fig. 1. In A. Data collection principle, two points would have suf- ficed, but the resolution of Thirty years ago this QuickTime® movies is limited, and film was available on a the high-speed camera did not regis- film loop, and students in ter frames very well, so an additional Project Physics8 were point was used as a cross check. We asked to view it carefully, made measurements every 10 frames, describe verbally what which is considerably more measure- happened as thoroughly ments than students would ever have as possible and to make a tried with pencil-and-ruler tech- simple measurement such niques. as checking to see We assumed that the stated frame whether momentum was rate (3000 fps) was accurate and that Fig. 2. Displacements of colliding billiard balls. Collision occurs conserved between the the film had been accurately trans- at t = 0. Raw data and least-squares fitted curves are shown. ␶ two balls. This could have ferred to videodisc. Sizes of billiard Skidding of cue ball stops at time c; skidding of target ball ␶ stops at T. been done by projecting balls for different games may vary single, stopped frames slightly. For calibration purposes, we and marking the ball used the following numbers supplied positions on a screen. with the film: Since the camera frame rate and diameter of a bil- mass = 170.3 g 1 liard ball are given quan- diameter = 5.24 cm = (2ᎏᎏ in) 16 tities, both linear and angular velocities can We also assumed each ball to be of then be roughly comput- uniform density so that its moment of ed. Later on, with the 2 inertia is I = ᎏᎏmR2. We imported the availability of video play- 5 ers, students viewing the raw data from VideoPoint into a clip could make marks on spreadsheet for analysis. Because a transparent sheet cover- image resolution was limited and ing a television screen because we measured only a few and then take measure- points per frame, the position and ments off the sheet. In angle measurements shown in Figs. 2 either of these situations, and 3 exhibit noticeable jitter. tracking the motion of the balls through many B. Analysis frames would have been We determined that, just before tedious and error prone. the collision, the cue ball was moving These days, a video clip with a velocity of 0.98 m/s and rotat- can be digitized, viewed ing with an angular velocity that is on a computer, and stud- consistent with no skidding. Fig. 3. Rotation angles of colliding billiard balls in Fig. 2. Raw ied in detail with software The general features of the motion data and least-squares fitted curves are shown. Skidding of cue such as VideoPoint.9 With are apparent in Fig. 2: the cue ball ␶ ␶ ball stops at time c; skidding of target ball stops at T. such software, the user moving with constant velocity, then can view the video clip halting suddenly and undergoing a one frame at a time and period of acceleration and finally

Reality and Theory in a Collision Vol. 37, Jan. 1999 THE PHYSICS TEACHER 25 achieving a constant velocity again; the system had been lost. the target ball being struck, starting 3. After the collision, 15% of the to move but decelerating as it skids rotational kinetic energy was across the surface and begins to lost. rotate, finally reaching a constant 4. Overall, 17% of the total kinetic velocity. In the film the target ball energy of the system was lost visually seems to rotate backwards during the collision. slightly after the collision, but the 5. During the skidding phase, pre- motion is so slight and happens so sumably the only force acting quickly that a viewer could easily on each ball is the friction dismiss it as an illusion. However, between it and the surface. Fig. 3 confirms the visual impres- Consequently we expected to sion. Such reverse rotation indi- see accelerations that were iden- cates that friction between real bil- tical in magnitude. Instead, the liard balls during their collision is acceleration of the cue ball measurable. (+3.13 m/s2) and the accelera- The post-collision motions are tion of target ball (–1.83 m/s2) divided into a skidding phase fol- were significantly different. The lowed by a rolling phase. We fitted Fig. 4. Displacements of colliding billiard balls simulated skidding times are different: ␶ the data from each phase to simple c by Interactive Physics. Initial velocity of cue ball = 1 m/s. = 90 msec, ␶ = 150 msec. curves using an iterative procedure. Frictional coefficients = 0.08. T From the graphs, we visually esti- 6. The final velocities of the cue mated the time at which skidding ball and the target ball were stopped by using the time-honored 0.251 m/s and 0.583 m/s respec- tradition of viewing the graphs at a tively, so their ratio is 0.43. In grazing angle. We then computed the simple model presented least-squares fits of the linear and above with an initial cue ball rotational motions to quadratic velocity of 0.98 m/s, the com- functions of t during the skidding puted values would be 0.28 m/s phase (constant acceleration) and to and 0.70 m/s for a ratio of 0.40. linear functions of t during the rolling phase (no acceleration). We Simulations then adjusted our estimate of the onset time of the rolling phase until Since the real data contain the computed velocities at the end significant noise, students might be of the skidding phase matched tempted to use a computer sim- those at the start of the rolling ulation program to try to understand phase. The skidding times, ␶, deter- what is happening. How well can a mined from linear motions or from simple simulation program like rotational motions could differ by Interactive Physics® reproduce the nearly 10%. This is one estimate of Fig. 5. Rotation angles of colliding billiard balls simulated data observed here? It is easy to set the precision of our collected data. by Interactive Physics. Initial velocity of cue ball = 1 m/s. up a head-on collision in The fitted curves have been super- Frictional coefficients = 0.08. See Fig. 4. Interactive Physics.10 The calcu- imposed on the data in Figs. 2 and lations in Interactive Physics are 3. done by numerical integration of If we assume that all the measure- about +3.13 m/s2 with a standard the equations of motion. In ments are normally distributed and error of about 10%. simulating collisions, accuracy is have the same standard deviation and Here is a list of the interesting achieved by limiting the amount of that the fit of the curve to the data is measurements we obtained. overlap of two colliding bodies (their good (as it seems to be to the eye), inter-penetration) and choosing then during the skidding phase of the 1. After the collision, 10% of the smaller time-steps for the integration. motion, the acceleration of the target linear momentum had been lost. You can choose the masses and ball (using 50 data points) is about (0.167 kg m/s versus 0.151 kg speeds of the balls colliding, the –1.83 m/s2 with a standard error of m/s). coefficient of restitution (elasticity about 5%, and the acceleration of the 2. After the collision, 17% of the parameter), and coefficients of cue ball (using 35 data points) is translational kinetic energy of friction. We chose the masses for the

26 THE PHYSICS TEACHER Vol. 37, Jan. 1999 Reality and Theory in a Collision simulation to be the same as the masses of actual billiard balls, set the initial velocity at 1 m/s, and set the elasticity parameter to 1, fully elastic collisions. We also specified that the objects not overlap by more than 0.1 mm. Choosing a smaller overlap made it too difficult to position the balls at the start of a simulation: they tended to bounce off the surface when released. Some shortcuts programmed into Interactive Physics limit its useful- ness in this situation. For instance, you assign each object its own coeffi- cient of friction and whenever two Conclusions References bodies interact, the program actually 1. Jearl Walker, Sci. Am. 249, 127 uses the smaller of the two friction Once again we find upon careful (1983). coefficients. In this situation, it examination that nature provides us 2. C. B. Daish, The Physics of means that you cannot have smaller with simple events that challenge our Ball Games (English U. P., frictional coefficients between the understanding. The qualitative fea- London, 1972). balls than between a ball and the sur- tures of a head-on billiard ball colli- 3. R. Evan Wallace and Michael face—as would be needed in order to sion are well predicted by our simple C. Schroeder, “Analysis of bil- replicate the physical situation. theories. However, when we look in liard ball collisions in two dimensions,” Am. J. Phys. Consequently, in order to model detail at the collision there are some 56(9), 815-819 (1988). the ball/ball interaction properly, we features that puzzle us, e.g., the rela- 4. George. Y. Onoda, “Comment had to use small coefficients of fric- tively large loss of energy for an on ‘Analysis of billiard ball tion everywhere so that the balls “elastic” collision and the difference collisions in two dimensions,’ would not interact too strongly with in post-collision accelerations. We by R. E. Wallace and M. C. each other, but this means that in the have not tried to film other collisions Schroeder,” Am. J. Phys. 57(5), simulation the balls skid for much between billiard balls to see if the 476-478 (1989). longer time than they did after the data are reproducible. We offer it as a 5. Jane Higginbottom Bayes and actual collision. Figures 4 and 5 show challenge to physics classes every- William T. Scott, “Billiard-ball the result of a simulation using a where: what is really happening collision experiment,” Am. J. coefficient of friction ␮ = 0.08. The here? Phys. 31, 197-200 (1963). 6. N. A. Kholodilin and R. F. basic qualitative features are the same We urge physics teachers to use Nagaev, “On the theory of col- as the actual collision; however, the analytical tools made possible by lision of billiard balls,” quantitatively, the final speeds of the computers to enable their students to Mechan. Solids 27(6), 44-50 balls are 0.229 m/s and 0.657 m/s, carefully study everyday events. Let (1992). that is, the cue ball is slower and the the students discover for themselves 7. “Physics: Cinema Classics,” a target ball faster than the actual the stimulating richness of nature’s set of videodiscs, (AAPT, experiment but both velocities are in mysteries. Let them have the satisfac- 1993) Disc 3, Side E, Chapter reasonable agreement with the simple tion of having wonderful ideas of 63. Available from Ztek Co., model presented earlier. Even though their own. 800-247-1603. the program’s elasticity parameter e 8. The Project Physics Course was set equal to 1, in the simulation Acknowledgments (Holt Rinehart and Winston, New York, 1970); Unit 3: “The more than 10% of the kinetic energy We would like to thank the U.S. Triumph of Mechanics.” of the cue ball was lost during the Air Force Academy for its hospitality 9. VideoPoint® is published by collision, so frictional effects during to visitors during a portion of this Lenox Softworks (www.lsw. the collision time seem to be impor- analysis. This paper is a completion com/videopoint) and distrib- tant in the simulation as they are in of work begun by Benjamin D. uted by PASCO scientific, 800- the actual experiment. Pippitt. 772-8700. 10. Interactive Physics® is avail- able from Knowledge Revo- lution, 800-766-6615.

Reality and Theory in a Collision Vol. 37, Jan. 1999 THE PHYSICS TEACHER 27