Derivation of velocity in nth Bohr orbit

[SucculentLamb]

Homework Statement

(a) Derive an expression for the electron's speed in the nth Bohr model.
(b) Prove that the orbit with highest speed is the n = 1 orbit,with v(1) = ke^2 / h
compare this with the speed of light , and comment on the validity of ignoring relativity (as we did) in discussing the hydrogen atom.
(c) the ratio
alpha = v(1)/c =ke^2 / hc
is called the " fine structure constant" and is generally quoted a s alpha ~ 1/37. verify this value

Homework Equations

So, I was thinking of using (momentum) mvr=hn/2(pi) which works out to give a nice expression for the velocity, but it fails in the next two parts of the problem.

I also looked at the equation for the bohr radius: a = hbar^2/ke^2m but again, I'm not sure that this would work with the rest of the problem.

The Attempt at a Solution

Like I said, I tried the above two equations, but neither seemed to yield an answer that would allow for solving of parts (b) and (c). Really, I would just love some help with part (a) to give me a push in the right direction. Thanks!

 

[ByronT]

We know

$$m \frac{v^2}{r} = \frac{k e^2}{r^2}$$

and

$$m v r = n \hbar$$ (the quantized momentum of the particle)

so

$$\frac{m v r}{r^2}v = \frac{k e^2}{r^2}.$$

That should give you enough info to solve for the electron's speed!

 

[SucculentLamb]

Thanks! That was very helpful!