A rocket moving in outer space with no gravity on a straight line has an instant

Question

A rocket moving in outer space with no gravity on a straight line has an instantaneous mass
m, and in a time dt ejects an amount of dm mass in its exhaust - which causes an increase in
its velocity by dv. Exhaust velocity is normally large and is denoted by u.
a) Why is the momentum conserved in this motion?
b) Starting with momentum conservation; p(at time t) = p(at time t+dt), show an
expression for the instantaneous velocity of the rocket.
c) If there were gravity, how would your expression be modified?

Explanation / Answer

(a) According Newton's third law of motion every action has equal and opposite reaction.

Consider rocket is moving along x direction as fuel burns it ejects mass dm in -x direction but at the same time velocity dv increases in x- direction. Change in momentum along -x direction is equal to change in momentum along x direction, so net linear momentum is null, that' why momentum is conserved. Since, momentum = mass × velocity

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