John Doe was reading several documents on rockets and the formula associated with their propulsion. He learned that rock

[answerhappygod]

John Doe was reading several documents on rockets and the
formula associated with their propulsion. He learned that rock
burning is onboard fuel while moving through space has velocity
v(t) and mass m(t) at time t. He further learned that if the
exhaust gases escape with velocity ve relative to
the rocket, it can be deduced from Newton's Second Law of Motion
that
mvd /dt =dm / dt ve
a) Based on this fact show that v(t) = v(0) - ln m(0)/m(t)
vs
b) John Doe was interested in solving a next question. For
the rocket to accelerate in a straight line from rest to twice the
speed of its own exhaust gases, what fraction of its initial mass
would the rocket have to burn as fuel? Find a solution to his
question.