Dynamics | Part IA, 2001

A spherical raindrop of radius a(t)>0a(t)>0 and density ρ\rho falls down at a velocity v(t)>0v(t)>0 through a fine stationary mist. As the raindrop falls its volume grows at the rate cπa2vc \pi a^{2} v with constant cc. The raindrop is subject to the gravitational force and a resistive force kρπa2v2-k \rho \pi a^{2} v^{2} with kk a positive constant. Show aa and vv satisfy

a˙=14cv,v˙=g34(c+k)v2a.\begin{aligned} &\dot{a}=\frac{1}{4} c v, \\ &\dot{v}=g-\frac{3}{4}(c+k) \frac{v^{2}}{a} . \end{aligned}

Find an expression for ddt(v2/a)\frac{d}{d t}\left(v^{2} / a\right), and deduce that as time increases v2/av^{2} / a tends to the constant value g/(78c+34k)g /\left(\frac{7}{8} c+\frac{3}{4} k\right), and thence the raindrop tends to a constant acceleration which is less than 17g\frac{1}{7} g.

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