Closest approach of alpha particle - two separate methods

[etotheipi]

Homework Statement

Find the distance of closest approach of an alpha particle and nucleus (of arbitrary atomic number Z)

Relevant Equations

$E_{k}$, relative velocities

I've found two methods for doing this problem and they give different answers.

Method 1: Assume the larger nucleus does not move, and simply equate energies before the collision and at the point of closest approach:$$\frac{1}{2} m v^{2} = \frac{qQ}{4\pi\epsilon_{0}r}$$

Method 2: Assume the larger nucleus is capable of moving; the distance of closest approach occurs when the relative velocity of the two particles is zero. From this, we can use CLM and then do another energy calculation:

Conservation of Linear Momentum:
$$m_{\alpha}v_{\alpha} = (m_{\alpha} + M_{nucleus})v_{common}$$Energy:
$$\frac{1}{2} m_{\alpha} v_{\alpha}^{2} = \frac{1}{2} (m_{\alpha} + M_{nucleus})v_{common}^{2} + \frac{qQ}{4\pi\epsilon_{0}r}$$It makes sense that the two methods give different answers for $r$, however this is a very common exam question, and it is never usually stated whether we permit the larger nucleus to move or not. So, I was wondering which approach I should use for this and similar problems. Thank you!

 

[PeroK]

Unless you can assume the nucleus is large enough that its movement is negligible you must use method 2. In the case where the nucleus is very large, methods 1 and 2 should give approximately the same answer.

 

[etotheipi]

Unless you can assume the nucleus is large enough that its movement is negligible you must use method 2. In the case where the nucleus is very large, methods 1 and 2 should give approximately the same answer.

Thank you, this is essentially what I was thinking. It can be quite hard though to determine whether the assumption is justified or not. For instance, in the one below, I'd assume since silver is about 30x as massive as helium we could get away with the simpler method.

I suppose the only way to get a better feel for when certain assumptions are justified is just by doing lots of practice problems.

 

[PeroK]

Here's an idea. In this collision, the energy gained by the large mass is approx $\frac{m}{M}$ of the energy of the system. Is that significant? In the above problem, this ratio is about $4/50$ = 8%.

So, do you need the answer to within 10%? If not, you can do a quick calculation by method 1.

 

[etotheipi]

Here's an idea. In this collision, the energy gained by the large mass is approx $\frac{m}{M}$ of the energy of the system. Is that significant? In the above problem, this ratio is about $4/50$ = 8%.

So, do you need the answer to within 10%? If not, you can do a quick calculation by method 1.

That's a nice rule of thumb!