# Distance of closest approach

1. May 3, 2014

### fayled

1. The problem statement, all variables and given/known data
An alpha particle is accelerated from rest through a potential dierence of 20 kV. It travels directly towards a stationary Beryllium nucleus (4 protons, 5 neutrons). Calculate the distance of closest approach.

The problem definitely wants a solution that takes into account the recoil of the stationary nucleus on approach.

2. Relevant equations
VCM=m1v1+m2v2/m1+m2
U=Q1Q2/4πε0d
T=mv2/2

3. The attempt at a solution
Let the oncoming alpha particle have speed v. VCM=4v/13. In the CM frame, we have a particle of mass 4m, charge 2e, speed 9v/13 heading towards on of mass 9m, charge 4e, speed -4v/13.

Initially, T=18mv2/3. This all becomes PE which will be 2e2/πε0d for closest approach d. Equating given d=e2/3πε0mv2. T=qV for a p.d V so v2=2qV/m. Then d=e/3πε0V. Plugging in the numbers given d=96fm.

I don't have a solution so would appreciate somebody having a check. I'm unsure about my approach because it assumes that the initial scenario (i.e as soon as the alpha particle has finished being accelerated through the p.d) has the PE as zero, which isn't realistic and the question doesn't indicate this is valid. Thanks.

2. May 3, 2014

### dauto

You went wrong when you said all the energy becomes PE. That's not the case. At closest approach both particles are moving at speed VCM

3. May 3, 2014

### fayled

But I'm working in the CM frame?

Last edited: May 3, 2014
4. May 3, 2014

### Redbelly98

Staff Emeritus
Fayled has transformed the problem into the CM frame, where VCM is zero. So the approach is correct.

But I did spot two mistakes or typos.
Looks good so far.

It should be "/13" at the end, not "/3". Do you agree?

Agree, once you fix the earlier mistake.

I agree.

I don't think that follows. Can you show the steps leading up to this expression (following the $v^2=\frac{2qV}{m}$ result)? What are you using for q?

It is safe to assume this is valid. The alpha should be a considerable distance from the berylium when it exits the accelerator, so there would be negligible PE at that time.

Last edited: May 3, 2014
5. May 3, 2014

### fayled

Right:

Yes I agree! In fact a follow up question of mine was going to be why doesn't TLAB=TTCM+T', with T' the KE of a particle of the total mass of the system moving at speed VCM? That error solves that (should have known better than thinking physics had broken and I hadn't made an error).

So T=18mv2/13

U=2e2/πε0d

d=13e2/9πε0mv2

Now v2=eV/m. Before I had v2=2qV/m. But I would then need q=2e,m=4m (i.e I should have used M say - I think this confused you).

d=13e/9πε0V=416fm (simply substituting the above expression into the one above it).

Hopefully that's right. Thanks for your help!

Last edited: May 3, 2014
6. May 3, 2014

### Redbelly98

Staff Emeritus
Looks good. You're welcome!

p.s. You're right, I was confused about m vs. 4m.

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