An experiment is performed in deep space with a system composed of two uniform

spheres, one with mass $50$ kg and the other with mass $100$ kg$.$ They have equal radii, r $=$

$0\mathrm{.}20$ m$.$ The spheres are released from rest with their centers $40$ m apart. Let the $50-$kg

sphere be initially at x $=0$ and the $100-$kg sphere be at x $=40$m$.$ They accelerate toward

each other because of their mutual gravitational attraction.

$($a$)$ What quantities are conserved for this system?

$($b$)$ When their centers are $20$ m apart, find the speed of each sphere.

$($c$)$ Where do the surfaces of the spheres collide?