Question

A spaceship with a mass of 4.60 10^4 kg is traveling at 6.30 10^3 m/s relative to a space station. What mass will the ship have after it fires its engines in order to reach a speed of 7.84 10^3 m/s? Assume an exhaust velocity of 4.86 10^3 m/s.

Answer 1

initial momentum = 4.6*10^4 * 6.3*10^3
= 29 * 10^7 kgm/s
no external forces so that is the final momentum

final momentum = ship mass m *7.84*10^3 + exhaust momentum

exhaust mass = (4.6*10^4 - m)

what is the exhaust speed?
I assume that 4.86 * 10^3 is relative to the space ship but we need it relative to the space station or ground.
the average speed of the space ship is
(7.84 + 6.30)10^3 /2 = 7.07*10^3
so the average speed of the exhaust = 7.07*10^3 - 4.86*10^3 = 2.21 *10^3
so
29*10^7=m*7.84*10^3+(4.6*10^4-m) *2.21*10^3

29*10^7=5.63*10^3 m +10.2*10^7
5.63*10^3 m = 18.8*10^7
m = 3.35*10^4

Answer 2

for a more elaborate analysis see
http://www.braeunig.us/space/propuls.htm
read starting around equation (1.11)