Understanding the coefficient of restitution (COR) using ass/spring systes Dr. David Kagan Departent of Physics California State University, Chico Chico, CA 9599-00 dkagan@csuchico.edu The coefficient of restitution (COR) for a solid object such as a baseball, colliding with a perfectly rigid wall can be defined as the ratio of the outgoing speed to the incoing speed. When a hardened steel ball bearing collides with a large hardened steel plate, the collision has a COR of close to one. On the other extree, when a foa (Nerf t ) ball collides with the sae plate, the collision has a COR of nearly zero. It sees to be generally the case that flexible objects ore often suffer low COR collisions, while rigid objects are ore likely to undergo higher COR collisions. The intent here is to explain this behavior with a ass/spring syste odel. COR and Elastic Collisions The COR is related to the conversion of the initial kinetic energy into internal energy during the collision. For the baseball collision described above, the COR would be one if all the initial kinetic energy were conserved and appeared as the kinetic energy of the outgoing ball. You could iagine the COR dropping as soe paraeter of the ball is varied and the collision repeated. As the COR decreases toward zero, a greater and greater fraction of the initial kinetic energy is converted to internal energy in the ball. So, the COR is one for elastic collisions and less than one for inelastic collisions. Webster s defines the elastic to ean, easily stretched or expanded, while synonys such as inflexible and unyielding are given for inelastic. It is quite ironic that elastic objects tend to experience inelastic collisions and inelastic objects are ore likely to undergo elastic collisions. In fact, it is possible that this disparity between coon language and technical jargon is a source of confusion for our students. The goal of this discussion is to understand conceptually why easily defored objects tend to have inelastic collisions while inflexible objects tend to have ore elastic collisions using the least coplex odel of a solid as possible. A Very Siple Model of a Solid Modeling a solid as a collection of asses and springs can have pedagogic benefits as students have a sense, perhaps fro cheical odels, that solids are coposed of atos (asses) held in place by electroagnetic forces (springs). Ganiel actually describes a lecture deonstration using a cart carrying asses and springs that illustrates where the issing kinetic energy goes in an inelastic collision. Zou 3 iproved the design of the cart and shared a series of guided-inquiry learning activities
for their use. Other authors have attepted to understand the transfer of echanical energy into internal energy using ass and spring odels 4,5,6. Reducing the ass and spring odel as far as possible leaves two asses,, connected by a spring of spring constant, k, as shown in Figure. The asses are infinitely rigid and have no internal structure, so they can t possess any internal energy. The internal energy in this syste is in the vibrations of the spring. Figure illustrates the collision of our object with a perfectly rigid wall, one that doesn t absorb any internal energy. The incoing speed of the object is. Both asses initially ove at this speed so the only energy in the object is the kinetic energy of these asses. The total energy of the syste is, K o = v o + v o = v o. () Figure : The colliding object is represented by two equal asses,, connected by a spring of spring constant, k. k k before v v k after Figure : The collision of the object with a perfectly rigid wall. After the collision, the ass that collides with the wall has a speed v and the other ass has a speed v. Now, the speed of the center of ass is v = (v + v ) and the kinetic energy of the center of ass is, K = ()v = 4 (v + v ). () This kinetic energy of the center of ass is equivalent to the acroscopic kinetic energy of the object. The reainder of the energy goes into the internal energy associated with the oscillations of the ass/spring syste, U int = " 4 (v + v ). (3) This energy is equivalent acroscopically to the internal theral energy of the object. If the acroscopic kinetic energy is conserved, then the collision is acroscopically elastic. That is,
U int = 0 " elastic collision (4) U int > 0 " inelastic collision. The COR of the collision is the ratio of the center of ass velocity after the collision to the center of ass velocity before, COR = v c, f = (v + v ) v (v c,i o + ) = v + v. (5) The COR can be related directly to the kinetic energies, K COR =. (6) K o Therefore, the COR is one for elastic collisions and as the COR gets saller, ore and ore of the initial kinetic energy is converted to the internal energy in the spring/ass syste resulting in a COR less than one. For the reainder of this discussion we will focus on the COR keeping in ind that it is a surrogate for the energy transferred to internal energy. A Naïve Model of a Collision In our very siple odel of a solid, the only available paraeter to vary is the spring constant. A siple thought is that a very large spring constant odels a very rigid object, while a very sall spring constant odels a very flexible object. We can then look for variations in the COR due to the rigidity of the siple solid. The ass/spring syste heads toward the wall as shown in Figure 3a. When the ass first collides with the wall, it instantaneously reverses direction and keeps the sae velocity, since neither the wall nor the ass can absorb any internal energy. v = 0 v = 0 before first collision after first collision spring copressed and asses at rest before second collision a b c d Now the two asses are heading toward each other at the sae speed with the uncopressed spring between the as in Figure 3b. Notice that the ass/spring syste has no center of ass velocity. Soon, both asses are at rest with the copressed spring between the as in Figure 3c. The asses now reverse direction and begin to speed up. All c after second collision Figure 3: Two collisions of the ass/spring syste with a perfectly rigid wall. Note that the center of ass reains fixed fro the tie of the first collision until the end of the second collision. e
the while, the center of ass has not oved. When the ass is just about to strike the wall again, it is traveling at the sae speed it had before the first collision. Meanwhile the other ass has the sae speed as before, but has reversed direction as shown in Figure 3d. The spring is now uncopressed. The ass collides with the wall and reverses direction again. Now, both asses are headed away fro the wall with the sae speed as they arrived and the spring is uncopressed between the as in Figure 3e. The COR for this collision appears to be zero after the first bounce off the wall because the center of ass of the syste in Figures 3b, 3c, and 3d is at rest and all of the energy is in the internal otion of the ass/spring syste. However, after the second bounce, the COR is one and the collision is elastic regardless of the spring constant. Our naïve idea the rigidity of an object can be odeled by varying the internal spring constant sees dooed 7. However, the ipulse exerted by the wall on the first ass during the collision is actually far ore coplex than the instantaneous ipulse we have assued. The devil is in the details of the collision which can be odeled using the speed of copressional waves that travel along the spring. Roura 8 presents a very clean conceptual discussion of this issue. Toward a Better Model A way to siplify the intricacies of the collision with the wall is to just use a second spring to oderate it. Cross 9 applies this odel to explain the faous basketball and tennis ball deo. In our case, two equal asses,, joined by an internal spring of spring constant, k, collide with a solid assive wall. The collision is ediated by a second external spring of spring constant, βk, which can only exert forces in copression and not in extension. The springs both have the sae natural length, a, as shown in Figure 4. Large values of β odel a solid with weak springs or one that is fairly flexible, while sall values of β suggest a solid with strong internal springs or one that is rather rigid. βk k initial later a a x x Figure 4: Two equal asses,, joined by a spring of spring constant, k, and natural length, a, collide with a solid assive wall. The collision is ediated by a second spring of spring constant, βk. The goal is to follow the otion of the syste until it stops interacting with the wall (x > a). Then find the center of ass speed of the syste to calculate the COR for the collision. Writing Newton s Second Law for each ass when x < a,
d x dt = "k(a # x ) # k[a # (x # x )] (7) d x = +k[a " (x dt " x )]. (8) Switching to the diensionless variables, u "# x a a a k t. (9) Noting that, x = a(" u ), (0) x = a(" u ) + x = a( " u " u ), () dx dt = dx d" du du dt d" = #a du k d" $ d x = #a k d u dt d" = #a k u %% () dx = dx dt dt " a du du = "a k dt d# " a du k d# = "a $ du k d# + du ' & ) % d# ( d x = "a k $ d u dt d# + d u ' & % d# ( $ & u ** + u ** ' ). % ( (3) The Second Law equations becoe, u "" + #u $ u = 0 and u "" + u "" + u = 0. (4) When the syste is not interacting with the wall (x > a), the Second Law equations are the sae except β = 0. Equations 7 and 8 are subject to the initial conditions, x (0) = a, x (0) = a, x (0) = ", and x (0) = ", (5) which eans, where, u (0) = 0, u (0) = 0, u "(0) = #, and u " (0) = 0, (6) " #. (7) ka You can solve these equations by the ethod of noral odes 0 if you are enaored with elegance, closed for solutions, and advanced atheatics. If you are less skilled and willing to tolerate nuerical round off errors, you can use a spreadsheet. I naturally chose the later. By varying the value of β, you can go fro a very gentle collision with a stout internal spring (sall β) to a very abrupt collision (large β) with a soft internal spring. The data fro the calculations are suarized in Table. The second colun is the COR after the first bounce. For large β, the COR is sall after the first bounce, exactly as described earlier for the ass/spring syste striking the rigid wall. After one bounce, all the energy is in the internal otion of the ass/spring syste as in Figures 3b, c, and d. The second colun is the COR after the second bounce. For large β, the COR is one after the second bounce, in agreeent with the earlier analysis of Figure 3e. The otion of each ass as a function of tie for β = 0 is shown in Figure 5. Notice that there is very little oscillatory otion as the ass/spring syste leaves the wall consistent with a COR near one. For sall β the COR is also equal to one. The case of β = 0. is also shown in Figure 5. Notice again there is alost no oscillatory otion as the ass/spring syste
exits. However, for interediate values of β the COR is noticeably less than one. In Figure 5 the graph for β= shown illustrating a odest aount of oscillatory otion after the interaction with the wall because the COR has now dropped below one. COR 0.98 0.96 0.94 0.9 0.9 0.88 0.86 0.84 0.8 COR vs. β 0.8 0. 0 00 β Figure 6: COR vs. β for the two ass and spring syste. β COR COR COR 00 0.005.000.000 50 0.00.000.000 0 0.05 0.997 0.997 0 0.05 0.990 0.990 5.0 0.9 0.965 0.965 4.5 0.38 0.957 0.957 4. 0.59 0.948 0.948 4.0 0.65 0.945 0.945 3.9 0.73 0.94 0.94 3.8 0.936 0.936 3.0 0.896 0.896.4 0.878 0.878. 0.877 0.877.0 0.88 0.88.0 0.974 0.974 0.5 0.997 0.997 0. 0.999 0.999 0. 0.999 0.999 Table : The COR as a function of β after each bounce and total. 4 x x 3 β = 0. 6 β = 5 4 3 β = 0 4 3 5 0 5 0 5 30 4 6 8 0 4 t 4 6 8 t Figure 5: Position vs. tie graphs for each ass. The blue line tracks the ass that interacts with the wall via the external spring while the red line follows the otion of the other ass. The horizontal line is at x = a where the external spring stops acting on the ass/spring syste. A graph of COR vs. β is the subject of Figure 6. While the COR is one for both large and sall values of β, it drops at interediate values with a iniu around β =. Once again our siple idea that the rigidity of an object can be odeled by varying the internal spring constant appears to be thwarted. The shape of Figure 6 suggests that we are looking at a resonance phenoenon. The internal spring has a natural oscillation as illustrated in Figures 3b, c, and d. Since
the c is fixed, the frequency will be the sae as that of a single ass oscillating on a spring of half the length and therefore, twice the spring constant so, " int = k. (8) The external spring, which has a spring constant of βk is acting on the syste which has a ass of. It has a natural frequency of, " ext = #k. (9) The ost energy will be coupled into the internal spring when it is driven at its natural frequency by the external spring, " ext = " int # $k = k # $ = 4. (0) Of course, you will iediately notice that the iniu is actually closer to β= than β=4. Looking at the graphs in Figure 5, you will see that it is only for low values of β that the otion of the external spring is close to siple haronic. For higher values of β the otion is ore coplex and therefore so is the force exerted by the external spring on the ass/spring syste. The ass/spring syste could be treated as a forced haronic oscillator. The Forced Haronic Oscillator To treat the ass/spring syste as a forced oscillator let s go back to the Second Law equations and replace the external spring with a tie dependent force, d x dt = F(t) " k[a " (x " x )] () d x = +k[a " (x dt " x )]. () It is now appropriate to switch to diensionless variables that represent the center of ass otion and the internal separation of the asses, y c " x +x a The Second Law equations becoe, " " where, Taking the su and difference, = x c a, y rel "# ( x # x a a ) =# $x a, and " # k t. (3) y c + " y rel = f (#) $ y rel and (4) y " c # " y rel = y rel, (5) f (") # ka F(t). (6) y " c = f (#) and (7) y " rel + y rel = f (#). (8) Equation 7 is siply a stateent that the force exerted by the wall, f ("), divided by the total ass,, is equal to the acceleration of the center of ass. Newton would be pleased. Equation 8 is the equation for the forced oscillation of our oscillator, which has a natural frequency of, consistent with Equation 8. The energy absorbed by the ass/spring syste during the collision will appear as oscillations at this frequency in the center of ass frae. The solution to the traditional forced oscillator proble is,
c n n$ " int #" ei" n n y rel = %, (9) where the Fourier coponent of f (") is given by, c n = " & f (#)e $i% n# d# and " " 0 n = #n $. (30) This is getting far too coplicated far too quickly, however, we do have a forced haronic oscillator. It is just that the forcing function has frequency coponents at any different frequencies and the aplitudes of these coponents are not as obvious as we had hoped when we developed Equation 0, which assued a forcing function at only one frequency. Putting together Equations 7,, and 6, F(t) = kaf (") = #k(a $ x ) % f (") = # $ x ( a ). (3) The forcing function doesn t just depend upon the external spring (β), but also on the position of the colliding ass. Due to the coupled nature of the equations of otion, the position of the colliding ass depends upon the internal spring as well as the external spring. Our two ass/one spring syste has only one natural frequency. Since the forcing function is a coposed of a collection of frequencies, if we allow our syste to have ore natural resonances, soething interesting is bound to happen. More Masses and Springs We can create ore natural resonances, if we add ore asses and springs to our syste. So, let the fun begin! Each tie we add a spring and a ass to the right hand side we add another coupled equation to the equations of otion. This creates an additional natural frequency, which could be found using the ethod of noral odes, and increases the coplexity of the forcing function. Since we don t actually need the noral ode frequencies anyway (we just need to understand that they exist) and we have been able to use a spreadsheet to this point, let s just add ore coluns and adly calculate. The result is the COR versus β curve shown in Figure 7. Each added natural frequency allows the ass/spring syste to absorb ore internal energy because of the added natural frequency. In the three ass syste you can begin to see that the COR for high β is not longer tending toward one. By the tie you get to four asses, the strongly suggestive single resonance shape of the two ass syste is entirely gone. At last, we begin to see the general trend we have longed for, rigid low β objects tending to have higher COR s than flexible high β objects.
Figure 7: COR versus b for asses (blue), 3 asses (red), 4 asses (yellow), and 5 asses (green). Higher Octaves In a sense, we have shown that an object in a collision is like a sophisticated bell, a bell that can produce any distinct tones, one for each natural resonance frequency. The external spring that excites the tones is really the bell itself. During the collision, the object is defored. This deforation causes copressions in its internal springs. As the collision proceeds, the internal springs can transit energy to other internal springs if the natural frequencies atch the excitation frequency reasonably well. The chances of a good atch depend upon the spacing of the natural frequencies. The closer they are together the better the chances of excitation. The spacing of the natural frequencies depends upon the stiffness of the object. As a basic exaple, recall 3 the natural oscillations of a string of length L. They are associated with wavelengths of, " n = L where n =,, 3, (3) n The natural frequencies are, f n = v " n = nv L, (33)
and the spacing between the are, (n +)v f n + " f n = " nv L L # $f = v L. (34) The speed of wave on a string is the square root of the ratio of the tension, F t, to the ass per unit length, µ, so, "f = F t L µ. (35) A ore taut string has a higher tension and so ore broadly spaced natural frequencies. The arguent presented here for a string holds ore broadly. Objects that are ore rigid have a greater spacing between their natural frequencies and therefore, these frequencies are less likely to get excited during a collision. It is harder to ring their bell. So as a general rule, stiffer ore rigid objects have a higher COR and ore flexible object have a lower COR.
Webster s New Collegiate Dictionary (G. & C. Merria Co., Springfield, MA, 974). U. Ganiel, Elastic and inelastic collisions: A odel, Phys. Teach. 30, 8-9 (99). 3 X. Zou, Making internal theral energy visible Phys. Teach. 4, 343-345(003). 4 N. D. Newby Jr., Linear collisions with haronic oscillator forces: The inverse scattering proble, A. J. Phys., 47 (), 6-65 (979). 5 J. M. Aguirregabiria, A. Hernandez, and M. Rivas, A siple odel for inelastic collisions, A. J. Phys., 76 (), 07-073 (008). 6 R. Gruebel, J. Dennis, and L. Choate, A variable coefficient of restitution experient on a linear air track, A. J. Phys., 39 (4), 447-449 (97). Actually this paper uses a pendulu to represent the internal energy. 7 Reference 4 uses the naïve odel and gets interesting results. However, instead of varying the spring constant, the ass of the non-colliding ball is varied. This is unsatisfying because this odel akes it appear as though the COR depends upon the ass of the colliding object. The sae holds for reference 5 where the COR depends upon the ass at the end of the pendulu that represents the internal energy. 8 P. Roura, Collision Duration in the Elastic Regie, Phys. Teach. 35, 435-436 (997). 9 R. Cross, Vertical bounce of two vertically aligned balls, A. J. Phys., 75 (), 009-06 (007). 0 F. S. Crawford, Waves Berkeley Physics Course v.3 (McGraw-Hill, New York, 968), pp. 0-54. M. L. Boas, Matheatical Methods in the Physical Sciences (John Wiley & Sons, New York, 966), pp. 348-350 Many authors have nice explanations of collisions viewed fro the perspective of the deforations during the collision. For exaple, D. Auerbach, Colliding rods: Dynaics and relevance to colliding balls, A. J. Phys., 6 (6), 5-55 (994). R. Cross, Differences between bouncing balls, springs, and rods, A. J. Phys., 76 (0), 908-94 (008). B. Leroy, Collision between two balls accopanied by deforation: A qualitative approach to Hertz s theory, A. J. Phys., 53 (4), 346-349 (985). 3 H. D. Young and R. A. Freean, University Physics (Pearson Addison-Wesley, San Francisco, 004), pp. 575-577.