A proton of mass mP and a positively charged nucleus of mass 6mP are widely sepa

Question

A proton of mass mP and a positively charged nucleus of mass 6mP are widely separated and traveling towards each other at the same speed of 2714 m/s along the same line. Because they are both positively charged, they repel each other and undergo a soft head-on collision. As you'll learn in Phys 126, the electrostatic repulsive force is a conservative force so the 'collision' is perfectly elastic when we consider the speeds a long time before and a long time after the collision (when they are essentially no longer interacting). What is the final speed of the proton (in m/s) when they are again widely separated? (i.e. when they exert negligible forces on each other, and their speeds are no longer changing.) (Note: you don't actually need the mass of the proton here, only the mass ratio '6', since mP will cancel out of both of your equations.)

Explanation / Answer

conservation of momentum
6 m 2714 - m 2714 = m x - 6m y

so 5*2714 = x - 6 y

y = (5*2714 -x)/6

conservation of energy
6 m 2714^2 + m 2714^2 = m x^2 +6m y^2

6 * 2714^2 + 2714^2 = x^2 + 6((5*2714 -x)/6)^2

x=6591.14 m/s

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