A ball of mass 0.25 kg is travelling with a velocity of 1.2 m/s when it collides with an identical, stationary ball. After the collision, the two balls move together with the same velocity. How fast are they moving?

This question involves the principle of the conservation of momentum. 

Recall: Assuming no external forces act, linear momentum is always conserved. This means the total linear momentum of two objects before they collide equals the total linear momentum after the collision. 

To start, let's work out the momentum of each ball separately. We'll call the moving ball "A" and the stationary ball "B". We will need the formula Momentum = Mass * Velocity, or p = m * v. 

For ball A: 

p = m * v

p = 0.25 * 1.2

p = 0.3 kg m s^-1

Let's call this value p_A. 

For ball B: 

p = m *v

p = 0.25 * 0

p = 0. 

Let's call this value p_B.

So now we can work out the total momentum before the collision: 

p_total = p_A + p_B = 0.3 + 0 = 0.3 kg m s^-1. 

This is where we bring in the principle of conservation of momentum. The question tells us that after the collision, the balls move together with the same velocity. This means we can treat them as one object, and use our formula to work out the velocity after the collision. 

We know that after the collision, the total momentum = 0.3 kg m/s, from the principle of conservation. 

We also know that the total mass of both balls (remember we're treating them as one object, so we can add the masses) is 0.5 kg. 

We simply plug these numbers into our formula for momentum: 

p = m * v

0.3 = 0.5 * v

v = 0.3/0.5

v = 0.6 m/s