Question

Two identical spherical asteroids (mass 3.07x10¹⁸ kg, radius 5.5x10³ m) are initially at rest, with their centers 6.01x10⁵ 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!

Answer 1

let
m = 3.07*10^18 kg
r = 5.5*10^3 m

initial distance between the asteroids, d1 = 6.01*10^5 m
final distance between the asteroids, d2 = 2*r

= 2*5.5*10^3

= 1.10*10^4 m

let v is the speed of asteroids when they collide.

Apply conservation of energy

KEf + PEf = KEi + PEi

2*(1/2)*m*v^2 - G*m^2/d2 = 0 - G*m^2/d1

m*v^2 = G*m^2*(1/d2 - 1/d1)

v^2 = G*m*(1/d2 - 1/d1)

v = sqrt(G*m*(1/d2 - 1/d1) )

= sqrt(6.67*10^-11*3.07*10^18*(1/(1.1*10^4) - 1/(6.01*10^5))

= 135 m/s <<<<<<<<<<----------------------Answer