**See this solution**

We have Lagrangian formula as :

where r and ϕ are the polar coordinates of the relative position between the **α**-particle and the proton and

is the reduced mass.

Now, obtain the Euler-Lagrange equations of motion and integrate them once, obtaining:

where E is the energy and

is the angular momentum. Both E and l are constants of the motion.

Now we use the data from the question. The energy is given by the sum of the kinetic energies of the particles at infinity:

where recall that the relative velocity is twice v .

Similarly, the angular momentum is given by

Now, we substitute these result in the equation we found above for the energy. The closest approach distance is given by the condition r=0 , since this is a turning point. In other words, the relative position decreases from infinity to D as the particles approach and then starts to increase again.

Therefore, we have, after some algebra:

In order for the x defined in the problem to appear, we must multiply by 64 and add 25. This leaves the result:

**Therefore, x=89**