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?
m1=62.9 kg
m2=697 kg
F=117 N
t=0.977 s.
v1 = velocity of astronaut after contact
v2 = velocity of satellite after contact
By the conservation of momentum,
m1 v1 + m2 v2 =0
v2=-(m1 v1)/m2 ....(1)
By the conservation of energies,
F*t = (m1/2)v1² + (m2/2)v2² ....(2)
eliminate v2 by substitution of (1) in (2):
F*t = (m1/2)v1² + (m2/2)(m1*v1/m2)²
Solve for v1=1.8 m/s
substitute in (1) to get v2=0.2 m/s.
From v1 and v2 calculate distance after specified time.
The two velocities are in opposite directions, so relative velocity is the sum of absolute values.
To find the distance between the astronaut and the satellite after 3.57 minutes, we need to calculate the displacement of both objects.
First, let's convert the time to seconds, as all the other measurements are already in metric units:
3.57 minutes = 3.57 * 60 = 214.2 seconds
To calculate the displacement, we need to determine the initial velocity (u) and final velocity (v) of both the astronaut and the satellite.
For the astronaut, we can use the formula: v = u + at
Where:
v = final velocity
u = initial velocity (which is 0 in this case, as the astronaut pushes off and starts with no initial velocity)
a = acceleration
t = time
The acceleration (a) can be calculated using the formula: a = F / m
Where:
F = force
m = mass of the astronaut
Substituting the given values, we have:
a = 117.0 N / 62.9 kg ≈ 1.861 m/s^2
Now, we can calculate the final velocity of the astronaut:
v = 0 + (1.861 m/s^2) * (0.977 s) ≈ 1.815 m/s
Next, let's calculate the displacement of the astronaut using the formula: s = ut + (1/2)at²
Where:
s = displacement
u = initial velocity
t = time
a = acceleration
Substituting the given values, we have:
s_astronaut = (0 m/s) * (214.2 s) + (1/2) * (1.861 m/s^2) * (214.2 s)^2 ≈ 0 + (0.5) * (1.861 m/s^2) * (45933.64 s^2) ≈ 85148.5 m
Now let's move on to the satellite.
Since no external force is acting on the satellite, its acceleration is 0.
Thus, the initial velocity equals the final velocity throughout.
For the satellite, the equation becomes:
s_satellite = v * t
Where:
s = displacement
v = final velocity (the same as the initial velocity in this case)
t = time
Substituting the given values, we have:
s_satellite = (1.815 m/s) * (214.2 s) ≈ 388.869 m
To find the distance between the astronaut and the satellite, we subtract the displacement of the satellite from the displacement of the astronaut:
distance = s_astronaut - s_satellite
distance = 85148.5 m - 388.869 m ≈ 84759.631 m
Therefore, after 3.57 minutes, the astronaut and the satellite are approximately 84759.631 meters apart.