Kvant Physics Problem 72
The solution correctly identifies that each ball undergoes uniform circular motion about its respective pivot, with angular velocities $\omega_1 = v/l$ and $\omega_2 = 2v/(2l) = v/l$, so the angular s…
Verified: no
Verdicts: FAIL + FAIL
Solve time: 15m07s
Source on kvant.digital
Problem
NOT FOUND
Checks
The solution correctly identifies that each ball undergoes uniform circular motion about its respective pivot, with angular velocities $\omega_1 = v/l$ and $\omega_2 = 2v/(2l) = v/l$, so the angular speeds are equal. The application of the principle that identical masses in a perfectly elastic head-on collision exchange velocities is correctly invoked, and the algebraic derivation of the first collision time $t_1 = \pi l / v$, the sequence of collision times $t_n = (2n-1)\pi l / v$, the time of the $k$-th collision $t_k = (2k-1)\pi l / v$, and the number of collisions within a time interval $N(t) = \left\lfloor \frac{1}{2}\left( \frac{vt}{\pi l}+1 \right) \right\rfloor$ is mathematically consistent. The units are correctly carried, as all expressions have dimensions of time or are dimensionless where appropriate. The limiting cases, such as $t < \pi l / v$ yielding zero collisions and the spacing between successive collisions being $2\pi l / v$, are internally consistent and physically reasonable.
However, the solution still relies on an implicit assumption regarding the initial positions and velocity directions of the balls. The problem states that the balls start at points $A$ and $B$ with velocities $v$ and $2v$ respectively, but the solution assumes these velocities are directed so that both balls reach the point of tangency simultaneously, ensuring periodic collisions. This assumption is not explicitly justified by a geometric analysis of the figure or by a derivation from the given initial conditions. Without this verification, the claim that the balls collide exactly at times $t_n = (2n-1)\pi l / v$ is not rigorously established, representing a critical justification gap.
Summary
The algebraic derivation of collision times and counts is correct under the assumption of synchronized initial phases and velocity directions. The physical principles are applied correctly, and the units and limits are consistent. The primary flaw is the unverified assumption regarding the initial alignment of the balls that guarantees the periodic collision sequence.
VERDICT: FAIL — the solution does not justify that the given initial positions and velocities produce the periodic collision times used in the derivation.