You are making a trip in a space shuttle, and your friend Jimmy is in another space shuttle. He shares the same circular orbit with you, 370 km above earth. However, he is 21 km ahead of you.
A) How long will it take you to catch up with your friend if you reduce your orbital radius by .9 km?
B) By how much must you reduce your orbital radius to catch up in 9 hours?
Thanks
To answer these questions, we need to understand the principles of orbital mechanics and calculate the necessary parameters.
A) To find out how long it will take you to catch up with your friend by reducing your orbital radius by 0.9 km, we can use the concept of relative velocity.
1. Calculate the relative orbital speed between you and your friend:
The orbital velocity around the Earth can be calculated using the formula: v = √(G * M / r), where G is the gravitational constant (6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)), M is the mass of the Earth (5.97219 x 10^24 kg), and r is the distance from the center of the Earth to your orbit (370 km).
v = √((6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)) * (5.97219 x 10^24 kg) / (370000 m))
≈ 7673 m/s
2. Calculate the time it takes for you to cover a distance of 21 km (the distance between you and your friend):
In general, the formula for calculating time is: time = distance / speed.
time = (21,000 m) / (7673 m/s)
≈ 2.74 seconds
Therefore, it will take you approximately 2.74 seconds to catch up with your friend by reducing your orbital radius by 0.9 km.
B) To determine the required reduction in orbital radius to catch up with your friend in 9 hours, we need to calculate the change in distance between your previous and new positions.
1. Calculate the relative orbital speed as before:
v = √((6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)) * (5.97219 x 10^24 kg) / (370000 m))
≈ 7673 m/s
2. Calculate the distance covered in 9 hours at the current orbital speed:
distance = speed * time
distance = (7673 m/s) * (9 hours * 3600 s/hour)
≈ 248,076,000 m
3. Calculate the radius reduction needed to cover the distance between you and your friend (21 km):
reduction = original distance - new distance
reduction = 21,000 m - 248,076,000 m
≈ -248,055,000 m
Therefore, to catch up with your friend in 9 hours, you need to reduce your orbital radius by approximately 248,055 km. Note that a negative value indicates moving towards the center of the Earth.