Technical
Inelastic collision and
the Hertz theory of impact
American Journal of Physics, Vol. 68, No. 10,
pp. 920-924, October 2000 ©2000 American Association of Physics Teachers. All rights reserved. ## D. Gugan## H. H. Wills Physics Laboratory, Royal Fort, Bristol BS8 1TL, United KingdomReceived: 19 May 1999; accepted: 2 December 1999 The area, ## Contents- I. INTRODUCTION
- II. EXPERIMENT
- III. RESULTS AND DISCUSSION
- IV. SUMMARY AND CONCLUSIONS
- ACKNOWLEDGMENTS
- REFERENCES
- TABLES
## I. INTRODUCTIONThe physics of collision has been studied at least since Newton's time. Much
of the interest has been in the longitudinal collision of Hertz calculated the deformation at contact between isotropic, homogeneous
bodies with spherical surfaces in the static, linear, elastic approximation
(see Love where where and The time taken to reach maximum compression is given by an elliptic integral
over the compression, and for an In practice, a more convenient parameter than where the power of 0.8 in the first of these expressions arises because the
potential energy of compression is proportional to The Hertz equations can be rearranged to give ratios of measured quantities
which should remain independent of the impact speed, and These are used later, as is also a ratio which is, in addition, independent of the elastic moduli involved in the collision, Hertz's theory can be tested by measurements of the functional form of The present results came from a student project on the coefficient of restitution
between croquet balls and mallets, made in order to establish the variations
which exist for typical equipment. The project included some measurements of
the area and duration of contact, but it was only much later realized that
these appear to agree with Hertz's theory of impact,despite the loss of about
40% of the kinetic energy of the collision. This unexpected agreement was surprising,
and it is discussed in what follows. ## II. EXPERIMENTA test anvil was made from a cylinder of steel weighing 70 kg (about 150 times
the mass of the ball) equipped so that mallet heads could be firmly clamped
to it. The coefficient of restitution,
The area of contact, The ball used was a Jaques "Eclipse" ball (commonly used in the game of croquet),
which fell on a wooden mallet. The ball is rather similar to a golf ball (but
bigger, with ## III. RESULTS AND DISCUSSION## A. Contact areas,
A |

The expressions for *K*(*U*) show that by making a small shift
in the value of *K*_{0}, and by scaling *U* in the ratio *U _{W}*/

where *X _{W}* and

The original aim of this work did not require the equipment used to be well
characterized, and no independent value exists for *X _{J}*.
For the wood, neither the sort used nor its grain alignment is known, but since
all woods have the same value of Young's modulus within a factor of 2 (Ref. 9)
(about the same factor as arises from accidents of growth), a value for

The results of Sec. III A show good agreement with Hertz's theory of impact, which is unexpected since about 40% of the initial kinetic energy is lost. The theory assumes that mechanical energy is conserved, which is clearly not true, so why do Hertz's equations appear to hold?

A considerable part of the initial kinetic energy does not contribute to the
kinetic energy of recoil, but is almost all ultimately dissipated by thermal
diffusion; this dissipated energy is not initially distinct from that which
is recovered, however, and one may expect that both fractions are identically
stored throughout the strain field. If this is correct, *then the elastic
potential energy function used by Hertz is valid for the initial compression,
and the calculated values of A and of* *T*/2 *are given correctly
using the initial impact speed U, even for inelastic collisions*. Of course,
the recoil speed is reduced after impact and Hertz's theory shows that the
subsequent separation must take longer than the initial compression by an amount
which can be calculated from the known values of *e*(*U*). However,
this systematic increase of* T* in the present experiments is only 0.018
ms for *each* of the speeds in Table I, an effect
at the limit of uncertainty (but see also Sec. III C 3)

The combination of measurements of *A*(*U*) and *T*(*U*) suggests
that the energy loss occurs *after* the time of maximum compression
of the ball, and this is confirmed by the recent experiments of Cross^{2} on
the mechanical hysteresis which gives rise to energy loss. Cross used a piezo-transducer
to obtain force-time data for a variety of bouncing balls, and obtained the
dynamic force-distance response by numerical integration. His results relate
to Hertz's theory in a number of ways, but the important point here is that *all* his
hysteresis curves are asymmetric, as shown schematically in Fig. 4 (cf.
his Fig. 2): The compression is at least approximately Hertzian,
but the force during recoil is notably depressed, and it is also clear that
the ball has not fully recovered its shape at separation. Cross also shows
quasistatic force-distance curves for the softer balls, using a 3-min strain
cycle, and even at this much slower strain rate, 10^{-4} of
that on impact, most balls show hysteresis and stress relaxation during unloading,
e.g., from the areas of Cross's hysteresis loops one finds that the relative
energy loss of the golf ball is17%
under quasistatic test (corresponding to *e* = 0.91), and 27%
under impact (i.e., *e* = 0.86; cf. the value of *e* = 0.844
at an impact speed of 1.47 m s^{-1} from Cross's Table I).

Cross did not study a croquet ball, but the ball studied here appears to be
similar in its behavior to his golf ball, with a compression stage which is
essentially elastic and Hertzian, and an energy loss which occurs mostly during
the hysteresis on recoil, and which includes energy carried off by the still
partly compressed ball. The energy loss of the ball measured here, *K*(*U*)
= *K*_{0} + *K*_{1}*U*^{0.4},
can now be understood as an irreducible part, *K*_{0}, arising
from the area under the quasistatic hysteresis loop, and a speed-dependent
part which increases as the impact time decreases and the hysteresis loop expands.
Using the measured value of *K*(*U*) for the intrinsic loss of
the Jaques ball, its value of *e* falls from 0.90 for very low impact
speeds to 0.82 for a speed of 1.47 m s^{-1}, values which are very
close to those given above for the golf ball studied by Cross.

The systematic *increase* of *T*(*U*) by 0.018 ms calculated
in Sec. III C follows from assuming that the ball recoils at reduced speed,
but in accord with Hertz's theory. This implies that the ball has regained
a strain-free, spherical shape, and carries off no elastically stored energy:
This is one possible model of collision; however, it is not compatible with
the mechanical hysteresis curves measured by Cross. The observed residual compression
at separation can be modelled by a different assumption, i.e., that the collision
is elastic and reversible up to the recoil speed *eU*, when separation
occurs (when the flux of elastic stored energy is too small to produce further
acceleration), while the ball carries off the rest of the energy, which is
ultimately dissipated. The Hertz equation for conservation of kinetic and elastic
stored energy can be solved to obtain the fraction of the maximum compression
which is retained at separation, *f *(= (1-*e*^{2})^{0.4}),
and also the resulting decrease of the collision time. For the speeds in Table
I, * f* varies from 63.3% to 66.8% when the *decrease* of
the collision times varies from 0.193 to 0.168 ms. The two models above represent
extremes of behavior, and the true correction for inelastic effects must lie
between their limits. In fact, the experimental value of *f* for Cross's
golf ball is about 16%, much lower than the 61% which can be calculated from
its value of *e*, and if this value of *f* is also appropriate
for the croquet ball, then the contact times of Table I will
all be *smaller* by 0.04
ms than predicted by Hertz's theory. A systematic deviation of 0.04 ms is at
only twice the level of the experimental uncertainty, and cannot be ruled out;
indeed there is some evidence for it, since if the values of *T* observed
here are all increased by 0.04 ms to correct for this effect, then the value
of the ratio (*A*/*UT*) becomes 49±2 mm, in *exact* agreement
with the prediction from Hertz's theory.

The object of this work was *not* to make a test of Hertz's theory
of collision, which can be regarded as proven for elastic impact^{6},
but to show that measurements on a croquet ball of contact area, *A*(*U*),
and contact time, *T*(*U*), are well described by his theory,
even though about 40% of the kinetic energy is lost on collision.

This result can be understood since the values of *A*, and *T*/2,
are largely determined bythe *compression* stage of impact, before appreciable
energy loss has occurred, and are thus given correctly by the Hertz theory.
The contribution to the total contact time from recoil needs correction, since
it is *increased* by a lower recoil speed, but is *decreased* by
a residual compression of the ball at separation. However, since *TU*^{-0.2} [cf.
Eq. (5)], both effects give rise to rather small corrections,
and Hertz's equations therefore continue to be a good approximation for inelastic
collision even when the energy loss is large. The two effects lie outside Hertz's
theory, so that the observed agreement with it is partly fortuitous, but many
inelastic materials will have mechanical hysteresis loops similar to that for
Cross's golf ball and the croquet ball studied here, so that the present initially
unexpected agreement with Hertz's elastic theory should in fact not be uncommon.
This remains to be seen.

These experiments began as undergraduate projects. I am grateful to several
students for their enthusiastic interest; in particular, to D. G. McDowall
for developing the timer, to him and J. M. Marshall for the contact time results,
and to A. Jennings and S. Tanner for the contact area results. I am also grateful
to Professor R. G. Chambers for helpful comments on an earlier draft of this
paper.

- J. F. Bell, "The experimental foundations of solid mechanics,"
*Handbuch der Physik*(Springer-Verlag, Berlin, 1973), Vol. VIa/1, especially pp. 313-331.

An interesting comparison of the (elastic) impact of cylinders and balls is made by D. Auerbach, "Colliding rods: Dynamics and relevance to colliding balls," Am. J. Phys.**62**, 522-525 (1994). First citation in article - R. Cross, "The bounce of a ball," Am.
J. Phys.
**67**, 222-227 (1999)

[cf. also his later work on more complicated collisions, "Impact of a ball with a bat or a racket,"**67**, 692-702 (1999)]. First citation in article - A. E. H. Love,
*A Treatise on the Mathematical Theory of Elasticity*(Dover, New York, 1944), 4th ed., pp. 193-200, 440. First citation in article - L. D. Landau and E. M. Lifshitz,
*Course of Theoretical Physics*(Pergamon, Oxford, 1959), Vol. 7, pp. 26-31. First citation in article - B. Leroy, "Collision between two balls accompanied by deformation: A qualitative
approach to Hertz's theory," Am. J. Phys.
**53**, 346-349 (1985). First citation in article - References to early work are given by Bell (Ref. 1)
and by Love (Ref. 3), but the best confirmation of Hertz's
theory known to the author is given by A. E.Kennelly and E. F. Northrup.

"On the duration of electrical contact between impacting spheres," J. Franklin Inst.**172**, 23-38 (1911). They were interested in short electrical pulses, not in testing Hertz's theory (which they do not mention), however their results agree well with his predictions for*T*(*U*). The best experiments, on identical steel balls in pendulum collision, give -0.205±0.005 for the exponent for speeds between 0.44 and 2.8 m s^{-1}(cf.-0.2), while the measured values of*T*agree with those calculated from the elastic 1%.

Kennelly and Northrup do not consider the possibility of inelastic behavior, but later work by J. P. Andrews, "Theory of collision of spheres of soft metals," Philos. Mag.**9**, 593-610 (1930), develops Hertz's theory to deal with deformation of the surface of the ball due to*plastic*flow on impact. Andrews' model leads to a decrease of the coefficient of restitution with impact speed, similar to results reported by Raman, but it does not appear to be generally applicable

[cf. J. P. Andrews, "Experiments on impact," Proc. Phys. Soc.London**43**, 8-17 (1931)]. First citation in article - W. A. Prowse, "The development of pressure waves during the longitudinal
impact of bars," Philos. Mag.
**22**, 209-239 (1936). First citation in article - P. G. Tait, "On impact,"
*Scientific Papers*(Cambridge U.P.,Cambridge, 1900), Vol. 2, pp. 221-279. First citation in article - G. W. C. Kaye and T. H. Laby,
*Tables of Physical Constants*(Longmans, London, 1995), 16th ed., p. 48. First citation in article - R. F. S. Hearmon, "The elastic constants of anisotropic materials," Rev.
Mod. Phys.
**18**, 409-436 (1946). First citation in article

Table I. Impact measurements
for the Jaques croquet ball on wood (and coefficient of restitution on steel)
at different speeds. The quantities in the three columns on the right are
constant according to Hertz's theory: A/UT is expected
to equal 1.068R, i.e., 49.1 mm;the other two columns depend on
the elastic constants of the surfaces, see the text. The values in parentheses
indicate the uncertainty in the preceding digit. |
||||||

Impact speed, U (m/s) |
Contact area, A (mm ^{2}) |
Contact time, T (ms) |
Coefficient of restitution, e, on steel |
A/UT (m) ×10 ^{3} |
A^{1/2}T^{2} (ms ^{2}) ×10 ^{10} |
A^{5/2}/U^{2} (m ^{3} s^{2}) ×10 ^{12} |

2.19 |
107 |
0.97 |
0.805 |
50(3) |
97(5) |
25(3) |

2.78 |
126 |
0.90 |
0.796 |
50(3) |
91(5) |
23(3) |

3.67 |
164 |
0.83 |
0.783 |
54(3) |
88(4) |
26(2) |

4.24 |
183 |
0.81 |
0.776 |
53(2) |
89(4) |
25(2) |

4.99 |
204 |
0.79 |
0.767 |
52(2) |
89(3) |
24(1) |

5.50 |
216 |
0.79 |
0.762 |
50(2) |
92(3) |
23(1) |

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