Approximately one billion years ago, the Moon orbited the Earth much closer than it does today. The radius of the orbit was only 24 400 km. The orbital period was only 23 400 s. Today, the average radius is 385 000 km; and the present period is 2.36 × 106 s. Assuming that the orbit of the Moon is circular, calculate the ratio of the speed of the Moon in its ancient orbit to the speed that it has today.
A) 15.8
B) 12.8
C) 10.2
D) 7.15
E) 6.39
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The speed of an object in a circular orbit is given by the formula v = 2πr/T, where r is the radius of the orbit and T is the period of the orbit.
The speed of the Moon in its ancient orbit is therefore v_ancient = 2π(24 400 km)/(23 400 s) = 6.57 km/s.
The speed of the Moon in its current orbit is v_current = 2π(385 000 km)/(2.36 × 10^6 s) = 1.03 km/s.
The ratio of the speed of the Moon in its ancient orbit to the speed that it has today is therefore v_ancient/v_current = 6.57 km/s / 1.03 km/s = 6.39.
So the answer is E) 6.39.
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