Question

2.) As a raindrop falls through a cloud, it collides with smaller droplets of mist and grows in mass (a) Derive a differential equation that relates the mass and velocity of the drop as it falls and accretes mass. Hint: Do NOT just differentiate d(mv)/dt, but start with the impulse-momentum theorem in differential form, like we did in the derivation of the rocket equation. Your "system" should include the raindrop itself and a small mass Δm of droplets with which the raindrop will collide in time Δ Consider the total momentum before and inmediately after the collision. After you take a limit Δ t → 0, this will yield a nonlinear differential equation for mass and velocity of the raindrop.' (b) The rate at which the drop accretes mass is proportional to the cross-sectional area and instanta- neous speed: dm/dt-Kr2u. Assume that the drops are constant density, so m-ρυ.4TI、3/3, where ρυ is the density of water. Show that the drop falls with constant acceleration a < g. Hint: The system of differential equations is really messy to solve. To make the math a little easier, you can assume from the beginning that v = at with a being constant. Also assume the initial mass and radius of the drop is zero. You should now manipulate the equations and solve for the constant a and show that it is a simple fraction of g that does not depend on any other constants (K or ρ. a result I find a little surprising.

Answer 1