Vertical motion of a spherical body with decreasing mass

Abstract

The paper studies the fall and ascent in air of a spherical body with radius linearly decreasing with time. The first integral of the nonlinear equation of motion is expressed in closed form in terms of Bessel functions on the assumption that the aerodynamic drag depends on the velocity as a quadratic polynomial. The second integral is evaluated by an approximate formula. The reliability of the analytic solution is confirmed by comparing it with a numerical solution of the Cauchy problem Keywords: spherical body, fall and ascent in air, vaporization, aerodynamic drag, Bessel function, Cauchy problem Introduction. Meshchersky [6] and Tsiolkovsky [11] are known to pioneer studies of the motion of bodies with variable mass. Their ideas were developed in rocketry, i.e., in the cases where a portion of the mass ejected from a moving body in a certain direction generates a propulsive force. Along with this, there are cases in nature where a moving body is losing mass in all directions. This is how the mass is decreased in falling burning meteorites, flying vaporizing droplets, etc. When a body loses its mass omnidirectionally, and with small relative velocity, the propulsive force is weak and can be neglected. However, a change in the size of the body affects the drag, which leads to variable coefficients in the equations of motion and complicates the theoretical study.There are other aspects that greatly complicate the study, even if the body moving through a fluid is of constant dimensions [13, 15], a spherical liquid droplet moving through a perfect compressible fluid undergoes periodic contractions and expansions [16], especially if resonances are possible [14].In particular, motion becomes nonstationary. Therefore, the concept of terminal velocity defined by Zhukovsky [4] makes no sense for a falling body with variable mass. The trajectory of a moving body with decreasing mass can terminate because of its complete combustion or evaporation, which is impossible if the mass of the flying body is constant. Thus, it makes sense to study the ballistics of a body with decreasing mass, even when there is no propulsive force.We will examine two cases of vertical motion (fall and ascend) of a spherical body with radius decreasing with time in a prescribed manner. The theory to be outlined can be used to calculate the height of a fire suppression unit above the potential burning surface in designing automatic fire-fighting systems. For effective delivery of dispersed fire-extinguishing fluids, a droplet must not vaporize more than half its size while flying to the fire front [1, 10]. Equation of Vertical Fall of a Spherical Body and Its Analytic Solution.Consider a falling spherical body with radius r varying as a linear function of time t: