Answer:
Mass needed to fire is [tex]3 \times 10^4 kg[/tex]
Explanation:
Mass of the spaceship [tex]m_{0} = 5.30 \times 10^4 kg[/tex]
Initial velocity of the spaceship[tex]v_{0} = 5.75 \times 10^3 m/s[/tex]
Final velocity of the spaceship [tex]v_{f} = 8.39 \times 10^3 m/s[/tex]
Take exhaust velocity [tex]u = 4.6\times 10^3 m/s[/tex]
The velocity of the spaceship in the space is
[tex]v_{f}=v_{0}+u\times ln(\frac{m_{0}}{m} )\\8.39 \times 10^3= 5.75 \times 10^3+ 4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\8.39 \times 10^3-5.75 \times 10^3= 4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\2.64 \times 10^3=4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\\frac{2.64 \times 10^3}{4.6 \times 10^3} = ln(\frac{5.30 \times 10^4}{m} )\\0.57=ln(\frac{5.30 \times 10^4}{m} )\\e^{0.57}=\frac{5.30 \times 10^4}{m}\\m=\frac{5.30 \times 10^4}{1.768} \\m=3 \times 10^4 kg[/tex]
Mass needed to fire is [tex]3 \times 10^4 kg[/tex]
