Answer:
The value is 
Explanation:
From the question we are told that
The initial velocity of the proton is 
At a distance R from the nucleus the velocity is 
The velocity considered is 
Generally considering from initial position to a position of distance R from the nucleus
Generally from the law of energy conservation we have that
Here 
is the change in kinetic energy from initial position to a position of distance R from the nucleus , this is mathematically represented as
=> 
=> 
=> 
And 
is the change in electric potential energy from initial position to a position of distance R from the nucleus , this is mathematically represented as
Here 
is zero because the electric potential energy at the initial stage is zero so
So
=> ![\frac{1}{2} * m *v_0^2 [ \frac{1}{4} -1 ] = k * \frac{q_1 * q_2 }{R}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%20%2A%20%20m%20%20%2Av_0%5E2%20%5B%20%5Cfrac%7B1%7D%7B4%7D%20-1%20%5D%20%20%3D%20%20%20k%20%20%2A%20%20%5Cfrac%7Bq_1%20%2A%20q_2%20%7D%7BR%7D)
=> 
Generally considering from initial position to a position of distance 
from the nucleus
Here represented the distance of the proton from the nucleus where the velocity is 
Generally from the law of energy conservation we have that
Here is the change in kinetic energy from initial position to a position of distance R from the nucleus , this is mathematically represented as
=> 
=> 
=> 
And is the change in electric potential energy from initial position to a position of distance from the nucleus , this is mathematically represented as
Here is zero because the electric potential energy at the initial stage is zero so
So
=> ![\frac{1}{2} * m *v_o^2 [-\frac{15}{16} ] = k * \frac{q_1 * q_2 }{R_f }](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%20%2A%20%20m%20%20%2Av_o%5E2%20%5B-%5Cfrac%7B15%7D%7B16%7D%20%5D%20%20%3D%20%20%20k%20%20%2A%20%20%5Cfrac%7Bq_1%20%2A%20q_2%20%7D%7BR_f%20%7D)
=> 
Divide equation 2 by equation 1
=> 
=> 
=>
