Problem 22.

Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. The first sphere has a diameter of 60 cm, a mass of  50 g, and contains   of charge. The second sphere has a diameter of 40 cm, a mass of  150 g and contains    of charge. Assume that no other forces are acting on them.

(a) Find the maximum speed achieved by the first sphere.

(b) Find the maximum acceleration achieved by the first sphere.

Solution

(a) The initial distance between the spheres (between the centers of the spheres) is 30 cm+20cm=50cm=0.5 m. The initial velocities of both spheres are zero. Then we release the spheres and due to repulsion between them they will move away from each other. Finally when the distance between them is very large (infinite) the velocities will be maximal.

To find these velocities we need to write down the energy and momentum conservation laws. If is the final velocity of the first sphere, is the final velocity of the second sphere then we have:

From the second equation we can find the speed of the second sphere, then substitute it in the first equation and find the speed of the first sphere:

Then

(b) The maximum acceleration is achieved when the distance between the spheres is the smallest. Then