A 65kg astronaut (including spacesuit and equipment), is floating at rest a distance of 6m from the spaceship when she runs out of oxygen and fuel to power her back to the spaceship. She removes her oxygen tank (3.0 kg) and flings it away from the ship at a speed of 15 m/s relative to the ship
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum of an isolated system before an event is equal to the total momentum after the event.
Here's how we can calculate the final velocity of the astronaut:
1. Calculate the initial momentum of the system (before the astronaut throws the oxygen tank):
Momentum = Mass × Velocity
Mass of the astronaut, spacesuit, and equipment = 65 kg
Initial velocity of the astronaut, relative to the spaceship = 0 m/s
Initial momentum of the astronaut = (65 kg) × (0 m/s) = 0 kg·m/s
2. Calculate the initial momentum of the oxygen tank:
Mass of the oxygen tank = 3.0 kg
Initial velocity of the oxygen tank, relative to the spaceship = 15 m/s
Initial momentum of the oxygen tank = (3.0 kg) × (15 m/s) = 45 kg·m/s
3. Calculate the final momentum of the system (after the astronaut throws the oxygen tank):
Final momentum of the system = 0 kg·m/s (conservation of momentum)
Final momentum of the oxygen tank + Final momentum of the astronaut = 0 kg·m/s
Final momentum of the oxygen tank = - Final momentum of the astronaut
4. Equate the momentum equations:
Final momentum of the oxygen tank = - Initial momentum of the astronaut
(3.0 kg) × (final velocity of the oxygen tank, relative to the spaceship) = - (65 kg) × (final velocity of the astronaut, relative to the spaceship)
5. Solve for the final velocity of the astronaut:
(3.0 kg) × (final velocity of the oxygen tank, relative to the spaceship) = - (65 kg) × (final velocity of the astronaut, relative to the spaceship)
Divide both sides of the equation by 65 kg:
(3.0 kg / 65 kg) × (final velocity of the oxygen tank, relative to the spaceship) = - (final velocity of the astronaut, relative to the spaceship)
Therefore, the final velocity of the astronaut is:
Final velocity of the astronaut, relative to the spaceship = - (3.0 kg / 65 kg) × (final velocity of the oxygen tank, relative to the spaceship)
Note that since the astronaut throws the oxygen tank away from the spaceship, the final velocity of the oxygen tank relative to the spaceship can be considered positive (in the direction away from the spaceship).