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Variable Mass Problem
By Edward Toman (P2478) on Monday, January 15, 2001 - 09:58 pm:

Could someone please answer the following question:

A raindrop falls through a stationary cloud. Its mass m increases by accretion uniformly with the distance x fallen, so that

m=m0(1+kx) [where m0 is the initial mass and k>0 is constant]

Given that its speed v is zero when x=0, show that

v2 = (2g/3k)(1 + kx - (1+kx)-2).

Thanks, Edward

[Editor: The following outline solution is courtesy of Carl James:

(1) Considering forces on the raindrop, Newton's 2nd Law yields d(mv)/dt = mg

(2) Write this as v2(dm/dx) + vm(dv/dx) = (d(mv)/dx)(dx/dt) = mg

(3) Substitute m expression in (2) to get, (1+kx)v(dv/dx) + v2k = (1+kx)g

(4) Multiply by 2(1+kx) so that, d[(1+kx)2v2]/dx = 2(1+kx)2g. Solve this differential equation, substituting in initial conditions.]