# A spaceship with a mass of 5.30 104 kg is traveling at 5.75 103 m/s relative to a space station. What mass will the ship have after it fires

Question

A spaceship with a mass of 5.30 104 kg is traveling at 5.75 103 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 8.39 103 m/s

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1 year 2021-09-04T09:47:04+00:00 1 Answers 7 views 0

Mass needed to fire is $$3 \times 10^4 kg$$

Explanation:

Mass of the spaceship $$m_{0} = 5.30 \times 10^4 kg$$

Initial velocity of the spaceship$$v_{0} = 5.75 \times 10^3 m/s$$

Final velocity of the spaceship $$v_{f} = 8.39 \times 10^3 m/s$$

Take exhaust velocity $$u = 4.6\times 10^3 m/s$$

The velocity of the spaceship in the  space is

$$v_{f}=v_{0}+u\times ln(\frac{m_{0}}{m} )\\8.39 \times 10^3= 5.75 \times 10^3+ 4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\8.39 \times 10^3-5.75 \times 10^3= 4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\2.64 \times 10^3=4.6 \times 10^3 \times ln(\frac{5.30 \times 10^4}{m} )\\\frac{2.64 \times 10^3}{4.6 \times 10^3} = ln(\frac{5.30 \times 10^4}{m} )\\0.57=ln(\frac{5.30 \times 10^4}{m} )\\e^{0.57}=\frac{5.30 \times 10^4}{m}\\m=\frac{5.30 \times 10^4}{1.768} \\m=3 \times 10^4 kg$$

Mass needed to fire is $$3 \times 10^4 kg$$