Space Chase: You have been sent in a small spacecraft to rendezvous with a space

Question

Space Chase: You have been sent in a small spacecraft to rendezvous with a space station in a circular orbit of radius r_0 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 circle from it! You do not apply forward thrust in an attempt to "chase" the station; that would be fatal folly. Instead, you apply brief braking thrust against the direction of your motion, to put you into an elliptical orbit, the highest point of which is your present position, but the period of which is half that of your present orbit. Thus you will return to your present position when the space station has come halfway around its circle to meet you. Is the minimum radius from the Earth's center-the low point-of your new elliptical orbit greater than the radius of the Earth, or have you botched your last physics problem? Do not guess!

Explanation / Answer

According to the question

We know that

mv^2/ro = GMm/ro^2

= v = sqrt(GM/ro)

Then

= Time period , T = (2 *pi * ro)/(sqrt(GM/ro))

= real time period of spacecraft = (pi * ro)/(sqrt(2GM/ro))

= Yes, the minimum radius from Earth's center of new elliptical orbit greater than the radius of earth .

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