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

Answer 1

Mass needed to fire is
`3 * 10^4 kg`

Step-by-step explanation:

Mass of the spaceship
`m_(0) = 5.30 * 10^4 kg`

Initial velocity of the spaceship
`v_(0) = 5.75 * 10^3 m/s`

Final velocity of the spaceship
`v_(f) = 8.39 * 10^3 m/s`

Take exhaust velocity
`u = 4.6* 10^3 m/s`

The velocity of the spaceship in the space is

`v_(f)=v_(0)+u* ln((m_(0))/(m) )\\8.39 * 10^3= 5.75 * 10^3+ 4.6 * 10^3 * ln((5.30 * 10^4)/(m) )\\8.39 * 10^3-5.75 * 10^3= 4.6 * 10^3 * ln((5.30 * 10^4)/(m) )\\2.64 * 10^3=4.6 * 10^3 * ln((5.30 * 10^4)/(m) )\\(2.64 * 10^3)/(4.6 * 10^3) = ln((5.30 * 10^4)/(m) )\\0.57=ln((5.30 * 10^4)/(m) )\\e^(0.57)=(5.30 * 10^4)/(m)\\m=(5.30 * 10^4)/(1.768) \\m=3 * 10^4 kg`

Mass needed to fire is
`3 * 10^4 kg`