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------------------------------------------------ An astronaut is on a 127 m lifeline circling the spaceship at 7 revolutions per minute. The length of the line is changed until his speed reaches 63 m/s. How long is the lifeline, in meters, at that time?

To find the length of the lifeline when the astronaut's speed reaches 63 m/s, we can use the formula for the circumference of a circle. Setting the linear speed to 63 m/s and solving for the circumference, the length of the lifeline is approximately 541.71 m.

Explanation:

To find the length of the lifeline when the astronaut's speed reaches 63 m/s, we can use the formula for the circumference of a circle:
Circumference = 2 * π * radius

Since the lifeline is in the shape of a circle, the length of the lifeline is equal to the circumference. We know the frequency of the astronaut's revolution, which is 7 rev per min. We can use this information to find the time taken to complete one revolution, which is 60 seconds divided by the frequency: Time taken for 1 revolution = 60 / 7 = 8.57 seconds

Now, we can use the formula for linear speed:
Linear speed = Circumference / Time

Setting the linear speed to 63 m/s and solving for the circumference, we have:
63 m/s = Circumference / 8.57 s

Simplifying this equation, the length of the lifeline is approximately 541.71 m.