Modelling Raindrops - Mass Increasing at Constant Rate

As raindrops fall, they tend to grow. If a raindrop of massis falling through a stationary cloud, then for each infinitesimally small amount that the raindrop grows by, momentum is conserved at the instant of growth. At the same time, gravity is pulling the raindrop down with a force Suppose the raindrop is accumulating mass at a constant rate C kg/s. If during a timethe momentum of the raindrop increases by

then assumingis constant during the time interval

Diving bygives

Now letto give the differential equation

If the mass of the raindrop is increasing at the constant rate C with respect to x thenandso

Write

The equation becomes

Now divide byand write:

We can solve this equation using the integrating factor method:

The equation now becomes

Hence

Ifwhenthen