An experiment is performed in deep space with two uniform spheres, one with mass

Question

An experiment is performed in deep space with two uniform spheres, one with mass 26.0 kg and the other with mass 107.0 kg . They have equal radii, r = 0.30 m . The spheres are released from rest with their centers a distance 41.0 m apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres.

A. When their centers are a distance 30.0 m apart, find the speed of the 26.0 kg sphere.

B. Find the speed of the sphere with mass 107.0 kg .

C. Find the magnitude of the relative velocity with which one sphere is approaching to the other.

D. How far from the initial position of the center of the 26.0 kg sphere do the surfaces of the two spheres collide?

Explanation / Answer

a)

v1 = initial velocity of sphere m1

v2 = initial velocity of sphere m2

m1 v1 = m2 v2

26 v1 = 107 v2

v2 = 0.243 v1                

r1 = initial distance between centers = 41 m

r2 = final distance between centers = 30 m

Using conservation of energy

change in gravitational potential energy = change in Kinetic energy

- (Gm1 m2) (-1/r2 - (-1/r1)) = (0.5) m v12 + (0.5) m v22

- (6.67 x 10-11(26 x 107)) (-1/30 - (-1/41)) = (0.5) 26 v12 + (0.5) 107 (0.243 v1)2

v1 = 0.0031 m/s

b)

v2 = 0.243 v1   = 0.243 x 0.0031 = 7.53 x 10-4 m/s

c)

relative speed = 0.0031 + 7.53 x 10-4 = 0.003853

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