letters to the editor 436 THE PHYSICS TEACHER ◆ Vol. 57, OCTOBER 2019 Analytic solution for a variable-mass snowball In the March 2019 issue, Scott Rubin analyzes the motion of a snowball rolling without slipping down a snowy inclined plane (making angle with the horizontal) and accreting ad- ditional snow as it does so. 1 Equations (12) and (13) of Rubin’s paper are two coupled differential equations for v and r, but they cannot be solved analytically in terms of elementary functions of independent variable t. Instead, Rubin performs a numerical integration in a spreadsheet to find v (t), along with the translational acceler- ation a(t) of the snowball’s center of mass. However, it is possible to find v and a analytically as a func- tion of a different independent variable, namely the snowball’s radius r. This result is arguably even more useful than know- ing how v and a vary with time, because we can directly relate r to the distance s the snowball has rolled down the slope as (1) using dr / d = k / 2π where is the angle through which the ball has rolled and r 0 is the snowball’s initial radius. Find v (r) as follows. Solve the right-hand part of Rubin’s Eq. (12) for v and substitute it into his Eq. (13) to obtain C – 60V 2 = 14rA (2) after simplifying, where V  dr/dt and A  dV/dt , and where C = 5 -1 gk sin is a constant. Now observe that (3) which is the same trick one uses to prove the work-energy theorem. Substitution of Eq. (3) into (2) results in an equation whose remaining variables r and V can be separated into (4) where the snowball initially has speed v = 0 and hence V = dr/dt must also be initially zero. After integrating, this result becomes , (5) yielding (6) Finally, a can be computed as (7) which has an initial value of (5/7) g sin and a terminal value of (1/6) g sin . Equations (6) and (7) are plotted in Fig. 1 and have a similar shape to Rubin’s graphs of v (t) and a(t) because V = dr/dt rapidly approaches a constant value of (12 ) –1/2 (gk sin ) 1/2 so that r can then be linearly rescaled as t. 1. S. Rubin, “A variable-mass snowball rolling down a snowy slope,” Phys. Teach. 57, 150–151 (March 2019). Carl E. Mungan U.S. Naval Academy, Annapolis, MD