Easy Conceptual Kinetic Energy Question: A conducting sphere of radius R is cent

Question

Easy Conceptual Kinetic Energy 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 = (2kq2 ) / 3R

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

The answer is Ki = (4kq2 ) / 3R, and I got this by doing the energy conservation equation Ki + Ui = Kf + Uf

However, the question had a hint stating that this question does not require any calculation (so I did not have to do Ki + Ui = Kf + Uf), so how do you get the answer conceptually without doing calculations?

Explanation / Answer

in the first case,

Ki + Ui = Kf + Uf

here, kf = 0

Ki + Ui = Uf

==> Uf - Ui = Ki

= (2/3)*k*q^2/R

in the seond case,

Ki = 3*(2/3)*k*q^2/R

= 2*k*q^2/R

change in potentail energy is same in both cases.

so, Uf - Ui = Ki

= (2/3)*k*q^2/R

Apply,

ki + Ui = Kf + Uf

==> Kf = Ki + Ui - Uf

= Ki - (Uf - Ui)

= 2*k*q^2/R - (2/3)*k*q^2/R

= (k*q^2/R)*(2 - 2/3)

= (k*q^2/R)*(4/3)

= 4*k*q^2/(3*R)

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