$(10\%)$ Problem $10$: A spherical rain drop of radius $R$ and mass $M$ falls vertically through a cloud layer. The drop enters the cloud layer at a height $H$ from the ground and exits the cloud at a height $h.$ While inside the cloud the drop accumulates water molecules so that its mass and size grow with time. The mass of the rain drop at time $t$ when inside the cloud is given as follows: $m(t)=M+at$ with $t=0$ at the time the drop enters the cloud, and $$ the rate at which water molecules accumulate.