Question

# Two identical spherical asteroids (mass 3.89x10^18 kg, radius 6.94x10^3 m) are initially at rest, with their...

Two identical spherical asteroids (mass 3.89x10^18 kg, radius 6.94x10^3 m) are initially at rest, with their centers 3.85x10^5 m apart. Ignore all other objects in the universe, and friction. The two asteroids accelerate toward each other, due to gravity, until they finally crash together. At what speed, in m/s, will each asteroid be moving when they collide (just touch on their outer edge)?

NOTE: when the asteroids collide, the distance between the centers of the asteroids is not zero!

Solution :

Given :

m = 3.89 x 1018 kg

R = 6.94 x 103 m

Initial separation (d1) = 3.85 x 105 m

Final separation (d2) = 2R = 2(6.94 x 103 m) = 13.88 x 103 m = 0.1388 x 105 m

According to Conservation of Energy : Change in KE = Change in PE

2 KEfinal = - G m2 / d1 - ( - G m2 / d2) = G m2 (1/d2 - 1/d1)

2 (1/2) m v2 = G m2 (1/d2 - 1/d1)

v2 = G m (1/d2 - 1/d1) = (6.674 x 10-11)(3.89 x 1018 kg){( 1 / 0.1388 x 105 m) - ( 1 / 3.85 x 105 m )} = 18030.176 m2/s2

v = 134.28 m/s

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