A 83.5 kg astronaut is working on the engines of a spaceship that is drifting through
space with a constant velocity. The astronaut
turns away to look at Earth and several seconds later is 35.5 m behind the ship, at rest
relative to the spaceship. The only way to return to the ship without a thruster is to throw
a wrench directly away from the ship. The wrench has a mass of 0.599 kg, and the astronaut throws the wrench with a speed of 21.6
m/s.
How long does it take the astronaut to reach
the ship?
Answer in units of s
To solve this problem, we need to use the concept of momentum.
The momentum of an object is given by the formula:
p = m * v
where p is the momentum, m is the mass of the object, and v is the velocity of the object.
Initially, both the astronaut and the wrench are at rest relative to the spaceship, so their initial velocities are zero.
The total momentum before the astronaut throws the wrench is zero, as there is no initial velocity:
0 = (mass of astronaut * velocity of astronaut) + (mass of wrench * velocity of wrench)
The astronaut throws the wrench with a speed of 21.6 m/s. Assuming the astronaut and wrench are the only objects in the system, the total momentum after the throw is given by:
final momentum = (mass of astronaut * velocity of astronaut) + (mass of wrench * velocity of wrench)
Since the final momentum is zero (because the astronaut returns to rest relative to the spaceship), we can write:
0 = (mass of astronaut * velocity of astronaut) + (mass of wrench * velocity of wrench)
Solving for the velocity of the astronaut:
(mass of astronaut * velocity of astronaut) = -(mass of wrench * velocity of wrench)
velocity of astronaut = -((mass of wrench * velocity of wrench)/mass of astronaut)
Substituting the given values:
velocity of astronaut = -((0.599 kg * 21.6 m/s) / 83.5 kg)
Now, to find the time it takes for the astronaut to reach the ship, we can use the equation:
velocity = displacement / time
Since the astronaut is 35.5 m behind the ship, the displacement is -35.5 m (negative because the astronaut is behind the ship). Rearranging the equation:
time = displacement / velocity
Substituting the values:
time = (-35.5 m) / velocity of astronaut
Now we can solve for the time.
Calculate his speed going back to the spacecraft after throwing the wrench, using conservation of momentum.
Then divide 35.5 m by that speed.