#### Releasing two charges

Two positively charged balls are brought together in a region of the universe where we can assume that nothing acts on them aside from one another. When they are released from rest, the balls accelerate away from one another. In case 1, what can you say about the balls?

1. They have the same mass.
2. They have the same charge.
3. Both of the above.
4. Neither of the above has to be true.

The two balls have equal-and-opposite velocities at all times, so their accelerations must also be equal-and-opposite. They experience equal-and-opposite forces, so their masses must be the same. The charge on each ball does not have to be the same. It's certainly the same sign, but even if the magnitudes are different the forces are still equal-and-opposite.

In case 2 the two balls have different masses. After the balls are released from rest, which ball has more kinetic energy?

1. The lighter one.
2. The heavier one.
3. They have equal kinetic energies.

When the balls are very far apart, what fraction of the original potential energy does the lighter ball have as its kinetic energy?

This is basically an elastic collision where momentum and energy are conserved. From the simulation we can see that after being released the lighter ball has four times the speed of the heavier one. To conserve momentum, the heavier ball must have a mass four times as large as the lighter one.
M = 4m

Conserving energy, we can say that:

Ui + Ki = Uf + Kf

where the initial kinetic energy is zero. This gives:

Kf = 1/2mv2 + 1/2MV2 = Ui - Uf

We can write this as 1/2(mv)v + 1/2(MV)V = Ui - Uf

but we know from momentum conservation that:
mv = MV and v = 4V

So, the lighter ball has four times as much kinetic energy as the heavier ball.

When the balls are very far apart their potential energy is close enough to zero that it is negligible. The original potential energy is now split between the balls. If the lighter ball has four times the kinetic energy, ir must have 80% of the original energy while the heavier ball has 20%.