A conducting sphere of radius R is centered at the origin. It carries a net char

Question

A conducting sphere of radius R is centered at the origin. It carries a net charge of +Q.

If a proton is projected in the -x direction from (+3R,0), the minimum kinetic energy it will need to reach the surface of the sphere is (initial kinetic energy) Ki = (2kq^2)/3R

Now, if the proton had triple this initial K, how much K (kinetic energy) will the proton have at x = R?

Apparently, the answer is Ki = (4kq^2)/3R, and you get this by multiplying (2kq^2)/3R by 2.

Why and how do you get this answer?

Explanation / Answer

Initial Kinetic Energy = 3*(2kq^2)/3R
Initial Potential Energy = k*q^2/3r

Final Kinetic Energy = ?
Final potenatal Energy =  k*q^2/r

Using Energy Conservation,
3*(2kq^2)/3R +  k*q^2/3R = K.Efin +  k*q^2/R
3*(2kq^2)/3R + k*q^2/R (1/3 - 1) = K.Efin
(2kq^2)/R - 2/3 * k*q^2/R = K.Efin
K.Efin =  k*q^2/R (2 - 2/3)
K.Efin = (4k*q^2)/3R

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