Modelling Raindrops - Mass Increasing at Constant Rate
As raindrops fall, they tend to grow. If a raindrop of mass
is 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 time
the momentum of the raindrop increases by
then assuming
is constant during the time interval
Diving bygives
Now let
to give the differential equation
If the mass of the raindrop is increasing at the constant rate C with respect to x then
and
so
Write
The equation becomes
Now divide by
and write
:
We can solve this equation using the integrating factor method:
The equation now becomes
Hence
If
when
then
