You have been sent in a small spacecraft to rendezvous with a space station that

Question

You have been sent in a small spacecraft to rendezvous with a space station that is in a circular orbit of radius 2.6700E4 km from the Earth's center. Due to a mishandling of units by a technician, you find yourself in the same orbit as the station but exactly halfway around the orbit from it! You do not apply forward thrust in an attempt to chase the station; that would be fatal folly. Instead, you apply a brief braking force against the direction of your motion, to put you into an elliptical orbit, whose highest point is your present position, and whose period is half that of your present orbit. Thus, you will return to your present position, when the space station has come halfway around the circle to meet you. Calculate the minimum radius from the Earth's center - the low point - of your new elliptical orbit. Is it greater than the radius of the Earth (6370 km), or have you botched your last physics problem?

Explanation / Answer

time period of circular orbit = 2pi*sqrt(r^3/GM) time period of elliptical orbit = 2pi*sqrt(a^3/GM) given, time period of circular orbit = 2*time period of elliptical orbit 2pi*sqrt(r^3/GM) = 2*2pi*sqrt(a^3/GM) => r^3 = 4a^3 a = r/(2^2/3) now, highest point = radius of apogee = r lowest point = radius of perigee = 2a - r = 2*(1/3)r - r = (2^1/3 - 1)r = (2^1/3 - 1)*(2.6700*10^4) = 0.69399 * 10^4 km = 6939.9 km no, it is more than the radius of earth. Looks like you made a right choice :P

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