by Let the mass of two balls be M and m, where M = (6 n )m for n N. The larger ball s rolled towards the lghter ball, whch ear a wall and ntally at rest. Fnd the number of collsons between the two balls untl the larger ball begns to move n the drecton opposte to ts ntal velocty. Assume all collsons are perfectly elastc.
Our Soluton Ths problem, unsurprsngly, begns wth conservaton of momentum and, due to the elastc nature of the collsons, conservaton of energy. In defnng we have x m M, () u n x = u n+ + x+ () u n + xs n = u n+ + xs n+, (3) where u n and are the velocty of the large and small ball, respectvely, after the nth collson. Do due the lnearty of momentum, we guess a lnear soluton of the form: u n+ = αu n + β + = γu n + δ, = un+ + α β un =, (4) γ δ where all α, β, γ, δ R are are constant. Substtutng (4) nto the momentum and energy expressons () and (3), respectvely, we have u n x = u n (α + xγ) + (β + xδ) u n + xs n = u n(α + xγ ) + u n (αβ + γδx) + s n(β + xδ ). By matchng coeffcents wth the constants wthn each expresson, we establsh the system: α + xγ = α + xγ = αβ + γδx = β + xδ = x β + xδ = x From here, we wsh to derve non-trval solutons for the constants α, β, γ, and δ, whch we clam are nonzero and n terms of x. Applyng ths restrcton and trudgng through the algebra, one obtans the unque soluton α = x + x, β = x + x, γ = + x, and δ = x + x..
These mply the transformaton matrx Λ = x +x +x x +x x +x, makng the recurrence n (4) become or more succnctly where un+ + = x +x +x x +x x +x v n+ = Λ v n, v n = un. un, (5) We now acknowledge that (5), though correct n applcaton, ot helpful n ts present form. We are certan that wth n = (essentally at t = ), v = Λ v as ths s the frst collson and so only one transformaton of Λ ecessary. Though for the next collson, Λ s surrepttously appled twce: v = Λ v = v = Λ(Λ v ) = Λ v. And ths s true n general: v n+ = Λ n+ v, promptng a more useful expresson of (5) to be un+ + = x +x +x x +x u n+ x s +x. (6) Ths expresson s ndcatve that the matrx Λ s lkely dagonalzable, meanng there may exst matrces D and P such that Λ = P DP λ, where D = and P = z λ z λ, λ where each z λ s a column and egenvector for the egenvalue λ of Λ. Notce, then, that f Λ s dagonalzable, Λ n+ = (P DP )(P DP ) (P DP ) = P D n+ P, (7) }{{} n+ factors 3
whch vastly smplfes the computaton. To proceed, we frst fnd the egenvalues and egenvectors of Λ. Settng cos θ = x + x = x = cos θ + cos θ, t follows from both the functonal expresson det(λ λi ) =, where I s the dentty matrx, and Euler s equaton e θ = cos θ + sn θ that λ = e ±θ = λ = e θ and λ = e θ. To compute the egenvectors z λ and z λ, we revert back to Λ n terms of x. Ths prompts the egenvectors x x z λ = and z λ =. Because the egenvalues of Λ are dstnct and the above egenvectors are lnearly ndependent, Λ s dagonalzable, thereby establshng the matrces P = x and P = x n the dagonalzaton of Λ. Furthermore, we have e θ e D = = D n+ (n+)θ =, e θ e (n+)θ makng Λ n+ = x e (n+)θ e (n+)θ by the logc n (7). Though t may seem abrupt, t s convenent for computatonal sake to now defne p as the number of collsons that occur untl the velocty vector of the larger ball egatve. Now, multplyng the above matrx expresson wth p a + and through rearrangements of Euler s equaton, we obtan Λ p = cos(θp) sn(θp) x sn(θp). cos(θp) For sake of brevty (and to not underwhelm the result of ths problem), we have excluded a majorty of the ntermedate algebra. 4
By (6), we have up s p = cos(θp) sn(θp) x sn(θp) u cos(θp) s As specfed n the problem statement, s = ; hence the matrx-vector product above becomes up s p = cos(θp), sn(θp) whch mples u p = cos(θp). By the way n whch we ve defned p (the number of collsons before the drecton of the velocty vector of the larger ball s reversed), t follows that cos(θp) and cos(θ(p+)) <, whch formulates the compound nequaltes cos(θp) > cos(θ(p + )) = θp π < θ(p + ), (8) because the cosne functon changes sgn at π. We are now nearly there. Recallng the defnton cos θ = x + x, t follows from the Taylor polynomals cos θ = n= ( ) n θ n (n)! that, to a second-order approxmaton n θ, and x + x = + ( ) n x n n= θ x = θ. By the defnton for x n () and the value of x n the problem statement, we have that θ n. Hence, substtutng nto (8), we obtan the nequalty p n π < p +. A compound nequalty of ths form s the defnton of the floor functon beng operated on p. Consequently, t follows that p = n π, suggestng that the number of collsons p that occur untl the larger ball commences to retreat s the nteger formed by the frst n + dgts of π. Qute spectacular! 5