How to Find Velocity of 10 MeV Alpha Particle for Scattering Calculation?

[SUDOnym]

Hello
I actually wanted to posted this in the "Homework" section but it is currently working for me..

The problem is:

Calculate the cross-section for the scattering of a 10 MeV alpha particle by a gold nucleus $$_{79}^{197}Au$$ through an angle greater than (a) 10 degrees (b) 20 degrees c 30 degrees.

My answer:

I know the relevant equation is:

$$\sigma=\pi(\frac{Zze^{2}}{4\pi\epsilon_{0}mv_{0}^{2}}\cot\frac{\theta}{2})^{2}$$

so its really just a plug and chug kind of problem... only thing I can't figure out is how to get the value for v_0... i know that is hidden somewhere in the fact that it is a 10 Mev particle and I assume E=mc^2 plays a roll too... but trying the following does not provide me with a useful value of v_o:

$$E=mc^{2}+\frac{1}{2}mv_{0}^{2}$$

neither does:

$$E=\frac{1}{2}mv_{0}^{2}$$

so my question is, how do I find the value of v_0 given that I know its an alpha particlee (so I know its mass) and also that I know its 10Mev particle?

Many Thanks!

 

[Drakkith]

What is "v_0" ?

 

[SUDOnym]

v_0 is the initial velocity that the alpha particle has before any interaction with the gold nucleus..

 

[nasu]

The correct relativistic relationship between energy and momentum is
E^2 = p^2 c^2 + m^2 c^4

Here m is the invariant mass (in the rest frame) and p =mv.
From this you can find p and then v.

 

[SUDOnym]

Thanks! to clarify though, the energy E, that I want to use will be the 10MeV...?

 

[nasu]

I think the 10 MeV should be just the kinetic energy. The rest energy of a proton is about 900 MeV so the 10 MeV cannot include the rest mass of the alpha particle.
The kinetic energy is
E^2 - m^2 c^4 = p^2 c^2

Actually at this energy the classic approximation should work quite well as KE<<E.

 

[SUDOnym]

thanks, nasu. E=1/2mv^2 was my initial try but I kept making arithmetic mistakes which is why I came on here... took me at least two hours to get the correct answer for the cross-section!

 

[xts]

it is a 10 Mev particle and I assume E=mc^2 plays a roll too...

Nope! Rest mass of $\alpha$-particle is 3.7GeV, so if you are speaking about 10MeV $\alpha$, it is its pure kinetic energy, and you have pretty non-relativistic case.