To calculate the amount of fuel needed, we can use the concept of conservation of momentum. The change in velocity can be achieved by expelling mass at a specific exhaust velocity.
First, let's calculate the initial momentum of the spaceship:
Initial momentum = mass * velocity
Initial momentum = 6000 kg * 500 m/s
Initial momentum = 3,000,000 kg·m/s
To achieve a final velocity of 600 m/s, the spaceship needs to increase its momentum by:
Change in momentum = mass * change in velocity
Substituting the given values:
Change in momentum = mass * (final velocity - initial velocity)
Change in momentum = 6000 kg * (600 m/s - 500 m/s)
Change in momentum = 6000 kg * 100 m/s
Change in momentum = 600,000 kg·m/s
Now, let's determine the amount of fuel needed by considering the exhaust velocity. The change in momentum is equal to the momentum gained by the expelled fuel:
Change in momentum = expelled mass * exhaust velocity
Substituting the given values:
600,000 kg·m/s = expelled mass * 1.5 km/s
To convert the exhaust velocity to m/s:
1.5 km/s = 1500 m/s
Now we can solve for the expelled mass:
expelled mass = change in momentum / exhaust velocity
expelled mass = 600,000 kg·m/s / 1500 m/s
expelled mass = 400 kg
Therefore, the spaceship would need 400 kg of fuel to increase its velocity from 500 m/s to 600 m/s with an exhaust velocity of 1.5 km/s.