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.

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

for a more elaborate analysis see

http://www.braeunig.us/space/propuls.htm
read starting around equation (1.11)

To calculate the final mass of the spaceship after firing its engines, we need to apply the concept of conservation of momentum.

The conservation of momentum states that the total momentum of a system remains constant in the absence of external forces. In this case, the system consists of the spaceship and the exhaust gases expelled by the engines.

The initial momentum of the spaceship can be calculated by multiplying its mass by its initial velocity. Therefore, the initial momentum is given by:
P_initial = m_initial * v_initial

Where:
P_initial: Initial momentum
m_initial: Initial mass of the spaceship
v_initial: Initial velocity of the spaceship

Similarly, the final momentum of the spaceship (after firing the engines) can be calculated by multiplying its final mass by its final velocity. Therefore, the final momentum is given by:
P_final = m_final * v_final

Where:
P_final: Final momentum
m_final: Final mass of the spaceship
v_final: Final velocity of the spaceship

Since the momentum is conserved, we have:
P_initial = P_final

Therefore, we can write the equation as:
m_initial * v_initial = m_final * v_final

Rearranging this equation, we can solve for the final mass:
m_final = (m_initial * v_initial) / v_final

Now, let's plug in the given values to calculate the final mass of the spaceship:

m_initial = 4.60 * 10^4 kg (given)
v_initial = 6.30 * 10^3 m/s (given)
v_final = 7.84 * 10^3 m/s (given)

m_final = (4.60 * 10^4 kg * 6.30 * 10^3 m/s) / (7.84 * 10^3 m/s)

By performing the calculation, we find m_final = 3.700255102 kg.

Therefore, the mass of the spaceship after firing its engines to reach a speed of 7.84 * 10^3 m/s will be approximately 3.70 kg.