Hertz elastic dynamics of two colliding elastic spheres

This paper revisits a classic problem in physics Hertz elastic dynamics of two colliding elastic spheres. This study obtains impact period in terms of hypergeometric function and successfully combines Deresiewicz’s three segmental solutions into one single solution. Our numerical investigation confirms that Deresiewicz’s inversion is a good approximation. As an essential part of this study, a general Maple code is provided.


I. INTRODUCTION
Body impact is a phenomenon of collision between bodies, in which the two colliding elastic spheres, as shown in Fig. 1, is of a popular one 1-7 . When bodies collide, they come together with some relative velocity at an initial point of contact. If it were not for the contact force that develops between them, the normal component of relative velocity would result in overlap or interference near the contact point and this interference would increase with time. This reaction force deforms the bodies into a compatible configuration in a common contact surface that envelopes the initial point of contact 6 .
In this short article, firstly we verify Deresiewicz's exact solution and find out its validation range. Then combine the three segmental solutions into one single solution. Once we have the single solution, we will propose an better approximate inversion solution.

II. EXACT SOLUTION
Introducing a transformation, ξ = kh 5/2 /(M v 2 ), with the help of symbolic software, Maple, we are able to find Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 September 2020 doi:10.20944/preprints202009.0151.v1 the solution of Eq.1 as follows or in terms of h, we have where hypergeom is the hyper-geometrical function. Hence, the duration of impact is given by T = 2t(1) = 4 5 B( 2 5 , 1 2 ) h0 v = 2.943275184 h0 v , which was obtained by Love 2 . The solution in Eq.4 is shown in Fig. 2 Similar to Deresiewicz 3 , we define a time ratio as follows The comparison of the time ratio is illustrated in both Table II and Fig. 3 below. From both Table II and Fig.3, it is delighted to see that the solution in Eq.4 and Eq.5 produce the same results as the three segmental solutions of Deresiewicz 3 , which reveals that the three segmental solutions in Eq.2 can be combined into a single solution as in Eq.4 and or Eq.5. In other words, the solution in Eq.4 is valid for the whole domain of ξ ∈ [0, 1].
However, since the solution in Eq.4 is monotonic, the solution only works in the domain of t ∈ [T /2, T ], and can not predict the value of t = t(ξ) in another half domain t ∈ [T /2, T ].

III. NUMERICAL SOLUTION AND MAPLE CODE
To get a solution for the full duration of impact, ie.,t ∈ [T /2, T ], numerical method must be adopted. For reader's ease teaching and research, a general Maple code is provided below.  A numerical example is shown in Fig.4 below.

IV. SOLUTION INVERSION
The solution in Eq.4 can be expanded in Taylor's series as follows t = h 0 v n 0 (2n)! (n + 2 5 )4 n (n!) 2 ξ n+ 2 5 . Generally speaking, the inversion ξ = ξ(t) is hard to be found. Therefore, Deresiewicz's approximation h D in Eq.3 is a remarkable inversion of the series in Eq.7. Although Deresiewicz mentioned that h D in Eq.3 was obtained from his data fitting, to get such elegant and accuracy expression, I guess that Deresiewicz might got some ideas from the Taylor'series of arcsin x = n 0 (2n)! (2n+1)4 n (n!) 2 x 2n+1 .

V. CONCLUSIONS
In conclusion, this short article successfully combined Deresiewicz's three segmental solutions into a single function, and shown that the Deresiewicz's inversion is a high accuracy approximation. A numerical example is carried out and a general Maple code is provided.