8.114 CALC A Variable-Mass Raindrop. In a rocket-propul- sion problem the mass is variable. Another such problem is a rain- drop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is dp dv dm Fext = + dt dt dt = Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m kx, where k is a constant, and dm/dt = kv. This gives, since Fext = mg, dv mg = m + v(kv) dt Or, dividing by k, dv xgx + v² dt This is a differential equation that has a solution of the form v = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the raindrop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113-117.)

8.114 CALC A Variable-Mass Raindrop. In a rocket-propul- sion problem the mass is variable. Another such problem is a rain- drop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is dp dv dm Fext = + dt dt dt = Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m kx, where k is a constant, and dm/dt = kv. This gives, since Fext = mg, dv mg = m + v(kv) dt Or, dividing by k, dv xgx + v² dt This is a differential equation that has a solution of the form v = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the raindrop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113-117.)

University Physics Volume 1

18th Edition

ISBN:9781938168277

Author:William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:William Moebs, Samuel J. Ling, Jeff Sanny

Chapter9: Linear Momentum And Collisions

Section: Chapter Questions

Problem 9.10CYU: Check Your Understanding Notice that the mass of the air in the tank was neglected in the analysis...

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