A softball of mass 0.220 kg that is moving with a speed of 5.5 m/s (in the positive direction) collides head-on and elastically with another ball initially at rest. Afterward it is found that the incoming ball has bounced backward with a speed of 3.9 m/s.

To solve this problem, we can use the principles of conservation of momentum and kinetic energy.

1. Conservation of momentum: In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision.

Let's denote the mass of the initially resting ball as m1 and the mass of the incoming ball as m2. The initial speed of the incoming ball is v1 (5.5 m/s), and after the collision, it bounces back with a speed of v2 (3.9 m/s).

Before the collision, the total momentum is given by:
Initial momentum = m1 * 0 (since the initially resting ball has no initial velocity) + m2 * v1

After the collision, the total momentum is given by:
Final momentum = m1 * v2 + m2 * (-v1) (negative sign indicates backward motion)

According to the conservation of momentum, the initial momentum and final momentum should be equal, thus:

m1 * 0 + m2 * v1 = m1 * v2 + m2 * (-v1)

2. Conservation of kinetic energy: In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Before the collision, the total kinetic energy is given by:
Initial kinetic energy = 0.5 * m1 * 0^2 (the initially resting ball has no initial velocity) + 0.5 * m2 * v1^2

After the collision, the total kinetic energy is given by:
Final kinetic energy = 0.5 * m1 * v2^2 + 0.5 * m2 * (-v1)^2 (negative sign indicates backward motion)

According to the conservation of kinetic energy, the initial kinetic energy and final kinetic energy should be equal, thus:

0.5 * m1 * 0^2 + 0.5 * m2 * v1^2 = 0.5 * m1 * v2^2 + 0.5 * m2 * (-v1)^2

Now, we can solve these two equations simultaneously to find the masses of the balls, m1 and m2.

It should be noted that this problem assumes an elastic collision, which means no energy is lost during the collision. Elastic collisions can occur between objects that deform and then return to their original shape, such as two rubber balls colliding.