A spacewalking astronaut with a mass of 78 kg pushes off a 540 kg satellite, exerting a 125 N force for the 0.5 seconds it takes him to straighten his arms. How far apart are the astronaut and satellite after 60 seconds?
To figure out the distance between the astronaut and the satellite after 60 seconds, we need to first determine the speed at which the astronaut pushes off the satellite. We can use Newton's second law of motion to calculate this.
The formula for force (F) is given by:
F = m * a
Where F is the force applied, m is the mass of the astronaut, and a is the acceleration.
In this case, the force exerted by the astronaut is given as 125 N, and the mass of the astronaut is 78 kg. So we can rewrite the equation as:
125 N = 78 kg * a
To find the acceleration (a), we rearrange the equation:
a = 125 N / 78 kg
a ≈ 1.60 m/s²
Now, using the formula for constant acceleration and time, we can calculate the speed (v) at which the astronaut pushes off the satellite:
v = a * t
Where v is the final velocity, a is the acceleration, and t is the time.
In this case, the acceleration (a) is 1.60 m/s², and the time (t) is 0.5 seconds:
v = 1.60 m/s² * 0.5 s
v = 0.80 m/s
Next, we can use the equation for distance (d) traveled during constant acceleration to find how far apart the astronaut and satellite are after 60 seconds:
d = (v^2 - u^2) / (2a)
Where d is the distance, v is the final velocity, u is the initial velocity (which is 0 in this case), and a is the acceleration.
In this case, the final velocity (v) is 0.80 m/s, the initial velocity (u) is 0, and the acceleration (a) is 1.60 m/s². The time (t) is 60 seconds:
d = (0.80 m/s)^2 / (2 * 1.60 m/s²) * 60 s
d ≈ 36 meters
Therefore, after 60 seconds, the astronaut and the satellite are approximately 36 meters apart.