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
 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
 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
 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
  from the nucleus
Here  represented the distance of the proton from the nucleus where the velocity is
 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
 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
 is the change in electric potential energy  from initial position to a  position of  distance  from the nucleus , this is mathematically represented as
  from the nucleus , this is mathematically represented as 
           
Here   is zero because the electric potential energy at the initial stage is  zero  so
 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 
               
=>           
 =>           
=>             