Bruce willis messed up and a small asteroid of mass m is about to collide with the earth.

as it enters the atmosphere with initial velocity v0 perpendicular to the atmosphere, it is subjected to the friction of air f=-λv. luckily for us it is also melting and its mass is given by m(t)=mo.(1-αt) (t=0 when it enters the atmosphere).

the question is to know if the asteroid will have sufficiently melted by the time it hits the earth! be careful: if m varies, f ≠ ma

1. determine the time dependence of the velocity of the asteroid as it falls through the atmosphere.

2. using a taylor expansion of v(t) close to t=0, explain in which cases the asteroid starts by accelerating or decelerating when it enters the atmosphere. (gravity is neglected because its contribution to the change in velocity as the asteroid falls through the atmosphere will be negligible given its initial velocity)