Analysis of the Motion of a Ball in the Barrel of a Musket.

摘要

The one-dimensional motion of a particle is analyzed when the force on it is inversely proportional to its displacement and directly proportional to the elapsed time. Such a force law describes a projectile in a musket barrel that is propelled by a hot ideal gas where either the number of moles or the temperature increases linearly with time due to the burning gunpowder. A particular solution to Newton's second law is found analytically for the case of zero initial position and velocity. For more general initial conditions, numerical integration is used to find the position of the particle as a function of time. A scaling argument shows that at long times, these numerical general solutions all converge to the analytic particular solution. Further analysis reveals how that convergence occurs: the general solutions slowly oscillate about the particular solution with a predictable period and amplitude. In addition to the dynamics, the energetics of the motion are analyzed.