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Problem 116 Easy Difficulty

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
$$F_{\mathrm{ext}}=\frac{d p}{d t}=m \frac{d v}{d t}+v \frac{d m}{d t}$$
Suppose the mass of the raindrop depends on the distance $x$ that it has fallen. Then $m=k x,$ where $k$ is a constant, and $d m / d t=k v$ . This gives, since $F_{\text { ext }}=m g .$
$$m g=m \frac{d v}{d t}+v(k v)$$
Or, dividing by $k$
$$x g=x \frac{d v}{d t}+v^{2}$$
This is a differential equation that has a solution of the form $v=a t,$ 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 $t=3.00 \mathrm{s}$ . (c) Given that $k=2.00 \mathrm{g} / \mathrm{m}$ , find the mass of the raindrop at $t=3.00 \mathrm{s}$ . For many more intriguing aspects of this problem, see $\mathbf{K} .$ S. Krane, Amer Jour. Phys, Vol. 49$(1981)$ pp. $113-117$

Answer

(a) $a=\frac{g}{3}$
(b) $\left.x\right|_{t=3}=14.7 \mathrm{m}$
(c) $\left.m\right|_{t=3}=29.4 \mathrm{g}$

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Video Transcript

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