A proton moves from a large distance with a speed u m/s directly towards a free proton originally at rest. Find the distance of closest of closest approach for the two protons in terms of mass of proton m and its charge e.

Explanation

As here the particle at rest is free to move, when one particle approaches the other, due to electrostatic repulsion other will also start moving and so the velocity of first particle will decrease while of other will increase and at closest approach both will move with same velocity. So if v is the common velocity of each particle at closest approach, then by' conservation of momentum' of the two protons system.
`m u = mv + mv` i.e., `v = (1)/(2) u`
And by conservation of energy'
`(1)/(2)m u^(2) = (1)/(2)mv^(2) + (1)/(2)mv^(2) + (1)/(4pi epsilon_(0)) (e^(2))/(r ) rArr`
`(1)/(2) m u^(2) - m((u)/(2))^(2) = (1)/(4pi epsilon_(0))(e^(2))/(r )` [as `v = (u)/(2)`]
`rArr (1)/(4)m u^(2) = (e^(2))/(4pi epsilon_(0) r)`
`rArr r = (e^(2))/(pi epsilon_(0) u^(2))`