A 62.9 kg spacewalking astronaut pushes off a 697.0 kg satellite, exerting a 117.0 N force for the 0.977 s it takes him to straighten his arms. How far apart are the astronaut and the satellite after 3.57 min?
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To find the distance between the astronaut and the satellite after 3.57 minutes, we need to calculate the displacement of the astronaut and the satellite separately, and then find the distance between them.
First, let's find the displacement of the astronaut:
We can use the formula for displacement (s) with constant acceleration:
s = ut + (1/2)at^2
Here, u is the initial velocity (which is 0 because the astronaut starts from rest), a is the acceleration, and t is the time.
Given that the time is 3.57 minutes, we need to convert it to seconds:
t = 3.57 min * 60 s/min = 214.2 s
Since the astronaut is in space and there is no gravitational force, the only force acting on the astronaut is from pushing off the satellite. Thus, the acceleration can be found using Newton's second law of motion:
F = ma
a = F/m
Given that the force exerted by the astronaut is 117.0 N and the mass of the astronaut is 62.9 kg, we can calculate the acceleration:
a = 117.0 N / 62.9 kg = 1.860 m/s^2
Substituting the values of u, a, and t into the formula for displacement:
s_astronaut = 0 * 214.2 + (1/2) * 1.860 * (214.2)^2
Now let's find the displacement of the satellite:
The force applied by the astronaut on the satellite causes an equal and opposite reaction force on the astronaut due to Newton's third law of motion. Thus, the astronaut and the satellite accelerate in opposite directions with the same magnitude.
Using Newton's second law with the given force of 117.0 N and the mass of the satellite of 697.0 kg:
a_satellite = F / m = 117.0 N / 697.0 kg = 0.1678 m/s^2
Substituting the values of u, a_satellite, and t into the formula for displacement:
s_satellite = 0 * 214.2 + (1/2) * 0.1678 * (214.2)^2
Finally, to find the distance between the astronaut and the satellite after 3.57 minutes, we subtract the displacement of the astronaut from the displacement of the satellite:
distance = |s_satellite - s_astronaut|
After calculating the displacement values, you can subtract them to get the distance between the astronaut and the satellite.