Question

Two spheres having masses M and 2M and radii R and 3R, respectively, are released from rest when the distance between their centers is 12R. How fast will each sphere be moving when they collide? Assume that the two spheres interact only with each other.

Answer 1

Answer:

$v_2 = \sqrt{\frac{GM}{3R}}$

$v_1 = 2\sqrt{\frac{GM}{3R}}$

Explanation:

As we know by energy conservation that change in gravitational potential energy of the system = change in kinetic energy of the two ball

So here we can say

$-\frac{GM(2M)}{12R} + 0 = -\frac{GM(2M)}{4R} + \frac{1}{2}Mv_1^2 + \frac{1}{2}(2M)v_2^2$

Also since there is no external force on the system of two masses so here total momentum of the two balls will remains conserved

$0 = Mv_1 + 2Mv_2$

$v_1 = -2v_2$

now we have

$\frac{GM^2}{2R} - \frac{GM^2}{6R} = \frac{1}{2}M(-2v_2)^2 + \frac{1}{2}(2M)v_2^2$

$\frac{GM^2}{3R} = Mv_2^2$

$v_2 = \sqrt{\frac{GM}{3R}}$

$v_1 = 2\sqrt{\frac{GM}{3R}}$