Let the closest distance upto which the alpha particle approaches be r.
At this distance the kinetic energy of the particle is zero.
Given: Initial kinetic energy of the particle, K.Ei=12mv2
Also initial potential energy of the system is assumed to be zero.
Atomic number of alpha particle, Zα=2
Final potential energy of the system at closest distance, P.Ef=K(Ze)2er
Using conservation of energy : K.Ei+P.Ei=K.Ef+P.Ef
∴ 12mv2+0=0+K2Ze2r
⟹ r=K4Ze2mv2∝1m (where, K=14πϵo)