5) Two heavenly bodies both with radii of 6×103 miles and masses of 7×1024 kg ar

Question

5) Two heavenly bodies both with radii of 6×103 miles and masses of 7×1024 kg are separated by 4×107 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.

Hint: Consider the initial "scene" before the planets gain any speed and the final scene just when the surfaces are about to collide. What types of energy are relevant in each scene? Are the centers of the planets still separate in the final scene? Will that separation (if it exists) play a significant role? ]

6) 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?

Hint: The lack of symmetry complicates this problem in an important way. In addition to energy conservation, is something else conserved that might help you deal with the asymmetry in the planet's masses and the effect on their motion?

Explanation / Answer

here,

5)

mass , m = 7 * 10^24 kg

radius , r= 6000 miles = 9.6 * 10^6 m

r = 4 * 10^7 miles = 6.4 * 10^10 m

let their final speeds be v

using conservation of energy

potnetial energy lost = kineti cenergy gained

G * m^2 * ( 1 /(2r)^2 - 1/(2r + d)^2) = 0.5 * 2 * m * v^2

6.67 * 10^-11 * 7 * 10^24 ( 1/( 2 * 9.6 * 10^6)^2 - 1/( (2 * 9.6 * 10^6 + 6.4 * 10^10)^2)) = v^2

solving for v

v = 1.13 m/s

the speeds of planets before they collide is 1.13 m/s

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