How Small Can Accelerators Detect Particles?

[crybllrd]

Homework Statement

the wavelength of the probing beam must be approximately the same as the size of objects we want to resolve. For each of the accelerators described below, determine the smallest size objects that are resolvable and name an object roughly that small, remembering that these are order-of-magnitude estimates.
(a) In the radioactive decay of radium, most emitted $$\alpha$$particles (4He nuclei) travel at 1.5 ×10^7 m/s. This is what was used in the Geiger-Marsden Experiment.
(there are actually four questions, but I am trying to make sure I am on the right track)

Homework Equations

p=$$\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

p=$$\frac{h}{\lambda}$$

$$\lambda$$=$$\frac{h}{p}$$

where h = Planck's constant

The Attempt at a Solution

Sorry, I am not too good with LATEX references...

a) v= 1.5e7 m/s
m=6.64e-27 kg

p=$$\frac{(6.64e-27 kg)(1.5e7 m/s)}{\sqrt{1-\frac{(1.5e7 m/s)^{2}}{3e8m/s^{2}}}}$$

I solve this equation to get momentum 9.97e20 kg*m/s, then plug it into the next formula to find the wavelength:
$$\lambda$$=$$\frac{6.626 J*s}{9.97e20 kg*m/s}$$

I get this answer:

$$\lambda$$=6.65e-55

which is much smaller than an electron, quark or a string.

Surely I did something wrong, but I can't find my mistake.

 

[kuruman]

Redo the momentum calculation. Your value of order 10²⁰ kg m/s is absurdly large for an alpha particle.

*** Additional comment on edit ***
Also your value of h is missing the appropriate powers of 10.

 

[crybllrd]

Thanks so much. I missed the math error, I didn't have the negative sign for the scientific notation.
I now have the answer of 6.65e-15m, which is about the size (as far as orders of magnitude go) of an electron.
That was the answer I was looking for!
Thanks again

 

[ideasrule]

Where does it say electrons are 10^-15 m across? Electrons, as far as we know, have 0 size. Experiments have certainly constrained its radius to no larger than 10^-18 m.

 

[crybllrd]

Where does it say electrons are 10^-15 m across?

In our homework notes, my professor left a link to
http://en.wikipedia.org/wiki/Orders_of_magnitude_(length)
That's where I got it.

 

[Redbelly98]

Note, that link actually says electrons are somewhere between 0 and 1e-15 meters.