Two hypothetical planets of masses m1 and m2 and radii r1 and r2, respectively,

Question

Two hypothetical planets of masses m1 and m2 and radii r1 and r2, respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. Note: Both the energy and momentum of the isolated two planet system are constant.
(a) When their center-to-center separation is d, find expressions for the speed of each planet and for their relative speed. (Use any variable or symbol stated above along with the following as necessary: G, the gravitational constant.)
v1 =
v2 =
vr =

(b) Find the kinetic energy of each planet just before they collide, taking m1 = 2.60 1024 kg, m2 = 7.00 1024 kg, r1 = 2.80 106 m, and r2 = 5.80 106 m.
K1 = J
K2 = J

Explanation / Answer

Since the masses are in a 1:5 ratio, and the total momentum remains zero, the large mass will have 1/5 the velocity of the smaller mass at all times. They collide when the separation distance is d = r1 + r2 = 5.6*10^6 m At that time, the total kinetic energy equals the potential energy loss, which is KE = G*M1*M2/d With 1/5 of the mass of M2, M1 will have 5 times the velocity and 5 times its kinetic energy, giving it 5/6 of the total KE. M2 will have the other 1/6.

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