CALC A Variable-Mass Raindrop.? In a rocket- propulsion problem the mass is variable. Another such problem is a raindrop 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 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, Or, dividing by k, 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 rain-drop 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.)
Solution 114CP We have to calculate the distance using the given problem. Now using this distance we can calculate the final mass of the raindrop. Step 1 We have We also know that So we have Using this values in the first equation we have Now also since we have We can write that Now equating the above two calculated x we can write that Step 2 2 So the acceleration of the raindrop is given by a = g/3 = 3.27 m/s . Hence the distance travelled in 3 second is given by So the raindrop will travel 14.7 m in 3 second.