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Two positive charges X and Y start far apart and at rest. X is given an initial velocity directed towards Y. Let p and E denote the total momentum and total energy of the system. Which statement is correct?

1. If Y is held fixed, both p and E are conserved.
2. If Y is held fixed, E is conserved but p is not.
3. If both are free to move, p is conserved but E is not.
4. If both are free to move, E is conserved but p is not.

Correct answer: If Y is held fixed, E is conserved but p is not.

Solution

If Y is fixed, an external (constraint) force acts on Y to keep it stationary, so the system's momentum is not conserved; however the electrostatic interaction is conservative so total mechanical energy is conserved. If both are free, the system is isolated, so both p and E are conserved. Thus the only correct single statement is: with Y fixed, E is conserved but p is not.

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