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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
