(a) Two protons are placed at separation `d` on a smooth surface. Find the speed of each proton when separation between them is doubled.
(b) An `alpha`-particle and a proton are placed at a separation `d` on a smooth surface.Find the speed of each when separation between them becomes infinite. (`m`: mass of proton , `e`: charge of proton.)

Explanation

(A) Since external force is zero , momentum of system is conserved . The speed of each proton will be same.
Energy conservation between `①` and `②`
`K_(1) + U_(1) = K_(2) + U_(2)`
`0 + (1)/( 4 pi in_(0)) .(e.e)/(d) = (1)/(2) mv^(2) + (1)/(2) mv^(2) + (1)/(4 pi in_(0)) .(e.e)/(2d)`
`mv^(2) = (1)/( 4 pi in_(0)) .(e^(2))/(2d) rArr v = sqrt((e^(2))/(8 pi in_(0) md))`
(b) By momentum conservation
`0 = - 4 mv_(alpha) + mv_(p) rArr v_(0) = (v_(p))/(4)`
By energy conservation
`K_(1) + U_(1) = K_(2) + U_(2)`
`0 + (1)/(4 pi in_(0)) . (2e .e)/(d) = (1)/(2) . 4 mv_(alpha)^(2)+ (1)/(2) .mv_(p)^(2) + 0`
`(e^(2))/(2 pi in_(0) d) = 2m ((v_(p))/(4))^(2) + (1)/(2) mv_(p)^(2) = (5)/(8) mv_(p)^(2)`
`v_(p) = sqrt((4e^(2))/(5 pi in_(0) md)) , v_(alpha) = (v_(p))/(4)`