A 64.7 kg astronaut is on a space walk away
from the shuttle when her tether line breaks.
She is able to throw her 14.1 kg oxygen tank
away from the shuttle with a speed of 7.45m/s
to propel herself back to the shuttle.
Assuming that she starts from rest (rela-
tive to the shuttle), determine the maximum
distance she can be from the craft when the
line breaks and still return within 68.2 s (the
amount of time she can hold her breath).
Answer in units of m
1.623
To determine the maximum distance the astronaut can be from the craft when the line breaks and still return within the given time, we can use the principle of conservation of momentum.
Step 1: Calculate the initial momentum of the astronaut and the oxygen tank:
Momentum = mass * velocity
Momentum of astronaut = 64.7 kg * 0 m/s (starting from rest) = 0 kg*m/s
Momentum of oxygen tank = 14.1 kg * 7.45 m/s = 104.945 kg*m/s
Step 2: Calculate the final momentum of the astronaut and the oxygen tank:
Since the astronaut is throwing the oxygen tank away, the final momentum of the oxygen tank will be zero (as it comes to rest). Therefore, only the momentum of the astronaut needs to be considered.
Final momentum of astronaut = momentum of oxygen tank
Step 3: Use the concept of impulse to relate the initial and final momentum with the force applied and the time taken.
Impulse = Force * time = change in momentum
The force applied on the astronaut is due to throwing the oxygen tank, but the force applied on the oxygen tank (that propels the astronaut backward) is equal in magnitude and opposite in direction. Therefore, the change in momentum of the astronaut is equal to the initial momentum of the oxygen tank.
Change in momentum = momentum of oxygen tank = 104.945 kg*m/s
Step 4: Use the impulse-momentum equation to determine the acceleration of the astronaut:
force * time = change in momentum
force = change in momentum / time
The force experienced by the astronaut is equal to the mass of the astronaut multiplied by its acceleration (F = ma):
ma = change in momentum / time
a = (change in momentum) / (mass of astronaut * time)
acceleration = 104.945 kg*m/s / (64.7 kg * 68.2 s)
Step 5: Use the kinematic equation to determine the maximum distance traveled by the astronaut:
In this case, the astronaut starts from rest and the only force acting on the astronaut is the force applied by throwing the oxygen tank. Therefore, we can use the constant acceleration kinematic equation:
distance = (initial velocity * time) + (1/2 * acceleration * time^2)
Since the astronaut starts from rest, the initial velocity is zero:
distance = (1/2 * acceleration * time^2)
Plug in the values:
distance = (1/2 * (104.945 kg*m/s / (64.7 kg * 68.2 s)) * (68.2 s)^2
Now we can calculate this to determine the maximum distance traveled by the astronaut.