Speed, momentum and total energy of a muon (SR)

[jasonchiang97]

Homework Statement

A muon has a mass of 106MeV/c². Calculate the speed, momentum and total energy of a 200MeV muon(a muon with a kinetic energy of 200MeV).

Homework Equations

E=γmc
K+mc²=E
γ=1/(1-β)^1/2
β=(v/c)²

The Attempt at a Solution

To solve for the speed I plugged E=γmc into K+mc²=E to obtain K/mc² + 1 = γ. Expanding gamma and rearranging I get 1-(K/mc²+1)^-1 = β. But plugging in the numbers on the left hand side gives me a zero as K<<c² and I think that I went wrong somewhere.

 

[TSny]

Homework Equations

E=γmc
K+mc²=E
γ=1/(1-β)^1/2
β=(v/c)²

The first, third, and fourth equations have errors.
[EDIT: Sorry. Your third and fourth equations are OK if you are defining the symbol β to equal (v/c)². But it is more common to use β for v/c.]

3. The Attempt at a Solution

To solve for the speed I plugged E=γmc into K+mc²=E to obtain K/mc² + 1 = γ.

E ≠ γmc. But your expression for γ looks correct.

Expanding gamma and rearranging I get 1-(K/mc²+1)^-1 = β.

Check your work here, I don't think you got the correct expression for $\beta$.

But plugging in the numbers on the left hand side gives me a zero as K<<c²

Did you mean to write K << mc²? If so, how did you come to this conclusion based on the information given in the problem?

 

[jasonchiang97]

Sorry line 1 was supposed to be E=γmc² and sorry I skipped so many steps. I'll write down everything I did.

K=mc²(γ-1)
K/mc² + 1 = 1/(√1-(v²/c²))
1-(v²/c²)=(K/mc² + 1)²
1-(K/mc² + 1)²=(v²/c²)
c²(1-(K/mc² + 1)²)=v²
√[c²(1-(K/mc² + 1)²)]=v

did I do anything wrong?

 

[jasonchiang97]

Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?

Well given the numbers, c² = 9 x 10¹⁶ and from the values of K and m given K/mc² = 0

 

[TSny]

Sorry line 1 was supposed to be E=γmc² and sorry I skipped so many steps. I'll write down everything I did.

K=mc²(γ-1)
K/mc² + 1 = 1/(√1-(v²/c²))
1-(v²/c²)=(K/mc² + 1)²
...
did I do anything wrong?

Check going from 2nd to 3rd line

 

[TSny]

Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?

Well given the numbers, c² = 9 x 10¹⁶ and from the values of K and m given K/mc² = 0

Note the units in the value of the mass, m =106 MeV/c².
So c² cancels in mc².

 

[jasonchiang97]

Sorry line 1 was supposed to be E=γmc² and sorry I skipped so many steps. I'll write down everything I did.

K=mc²(γ-1)
K/mc² + 1 = 1/(√1-(v²/c²))
1-(v²/c²)=(K/mc² + 1)^-2
1-(K/mc² + 1)^-2=(v²/c²)
c²(1-(K/mc² + 1)^-2)=v²
√[c²(1-(K/mc² + 1)^-2)]=v

did I do anything wrong?

I keep making mistakes as I try to type it on the site sorry and thanks.

 

[jasonchiang97]

Note the units in the value of the mass, m =106 MeV/c².
So c² cancels in mc².

Yea I know they cancel I mean when you put in the numbers you basically get 1/∞ for E/mc²

 

[TSny]

What numerical value do you get for mc² (including units)?

(Your final expression for v in post #7 looks correct.)

 

[jasonchiang97]

What numerical value do you get for mc² (including units)?

(Your final expression for v in post #7 looks correct.)

ohh I see the c² cancels out. Thanks!