Question

Two heavenly bodies both with radii of 7×10³ miles and masses of 9×10²⁴ kg are separated by 2×10⁷ miles. They are initially at rest. How fast are they moving just before their surfaces collide. Assume that you can ignore any effects having to do with the existence of atmospheres and that nothing significant exists in the space between the planets.

Consider the same setup as before, only one planet is twice the mass of the other (their radii are still the same size). What is the speed of the more massive planet just before their surfaces collide?

Answer 1

Applying energy conservation,

PEi + KEi = PEf + KEF

-GM M / r + 0 = - G M M / (R + R) + 2 ( M v^2 / 2)

G M / 2 R - G M / r= v^2

v^2 = (6.67 x 10^-11 x 9 x 10^24)[1/(2 x 7 x 10^3 x 1609 m) - 1/(2 x 10^7 x 1609m)]

v = 5160 m/s .......Ans

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Applying momentum conservation,

0 = m v - 2 m V

v = 2 V

energy conservation,

-GM M / r + 0 = - G M M / (R + R) + (M v^2 /2 ) + (2 M V^2 / 2)

G M / 2 R - G M / r= (4 V^2 / 2) + (V^2) = 3 V^2

3v^2 = (6.67 x 10^-11 x 9 x 10^24)[1/(2 x 7 x 10^3 x 1609 m) - 1/(2 x 10^7 x 1609m)]

v = 2979 m/s .......Ans