Thread: gravity between 2 objects

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.

Originally Posted by gracy

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.

There are a number of tackling this. One of the simples utilises the conservation of energy and of linear momentum.

The two spheres start at a seperation of 12R, and end at a seperation of 4R,
so the change in potential energy of the system is equal to the work required
to take the spheres from a seperation of 4R to one of 12R:

DE=int(r=4R:12R) G _1 m_2/r^2 dr = G m_1 m_2/(6R)

So as the spheres start from rest this is equal to the sum of their KE's
when they hit:

G m_1 m_2/(6R) = (m_1 v_1^2 + m_2 v_2^2)/2.

The conservation of momentum gives:

m_1 v_1 = -m_2 v_2.

Solve these two equations to find v_1 and v_2.

RonL