Decay of Mass M Moving East to Photons: V/C = [sqrt(5)-1]/2

[mathman44]

Homework Statement

A mass M moving at speed V going east decays into 2 photons. One moves perpendicular (south) and the other at some angle from the horizontal. Show that if tan(theta) = 1/2, then V/C = [sqrt(5)-1] / 2

The Attempt at a Solution

(gamma is the lorentz factor)

Momentum in x: x momentum of angled photon: gamma*m*v
Momentum in y: y momentum of angled photon = momentum of perpendicular photon = E1/C

Solving for E1, I get mc^2/2*gamma

Since tan(theta) = 1/2 = [P in y]/[P in x]

then 1/2 = [E1/C]/[gamma*m*v]

I get V = C/gamma^2...

 

[tiny-tim]

Homework Statement

A mass M moving at speed V going east decays into 2 photons. One moves perpendicular (south) and the other at some angle from the horizontal. Show that if tan(theta) = 1/2, then V/C = [sqrt(5)-1] / 2

The Attempt at a Solution

(gamma is the lorentz factor)

Momentum in x: x momentum of angled photon: gamma*m*v
Momentum in y: y momentum of angled photon = momentum of perpendicular photon = E1/C

Solving for E1, I get mc^2/2*gamma

Since tan(theta) = 1/2 = [P in y]/[P in x]

then 1/2 = [E1/C]/[gamma*m*v]

I get V = C/gamma^2...

Hi mathman44!

Why gamma?

Photons travel at the speed of light.

 

[mathman44]

Hi mathman44!

Why gamma?

Photons travel at the speed of light.

Conservation of momentum says that the x momentum of the angled photon is equal to the x momentum of the mass before decay... right? So to find the momentum of the mass before decay, I need gamma.

 

[Redbelly98]

The Attempt at a Solution

(gamma is the lorentz factor)

Momentum in x: x momentum of angled photon: gamma*m*v
Momentum in y: y momentum of angled photon = momentum of perpendicular photon = E1/C

Solving for E1, I get mc^2/2*gamma

Since tan(theta) = 1/2 = [P in y]/[P in x]

then 1/2 = [E1/C]/[gamma*m*v]

I get V = C/gamma^2...

Looks good so far. What is gamma, by definition?

 

[mathman44]

Looks good so far. What is gamma, by definition?

gamma = 1/sqrt(1 - v^2/c^2)

:S?

 

[tiny-tim]

oh i see, you're using gamma for the mass M …

but you need the momentum of both photons, and you also need conservation of energy.

 

[mathman44]

I used cons momentum with both photons, no?

y: momentum of photon 1 = momentum of photon 2
x: momentum of photon 2 = momentum of mass before decay

where photon 2 is the angled one. I have already solved for E1, so what do I need cons. of energy for?

 

[Redbelly98]

I get V = C/gamma^2...

gamma = 1/sqrt(1 - v^2/c^2)

You can combine these equations to find v

 

[mathman44]

You can combine these equations to find v

Then V = C - V^2/C? Doesn't make sense to me...

 

[tiny-tim]

Then V = C - V^2/C? Doesn't make sense to me...

actually, V²/c = 1 - V²/c, but still gives the wrong answer

… I have already solved for E1, so what do I need cons. of energy for?

mathman44, i don't know how you solved for E1, but you do need cons. of energy

 

[mathman44]

Yes, you're right... my teacher gave us E1 in class. Does

gamma*m*c^2 = p1c + p2c

look good? And if so, I don't know p2 so how can I solve for p1c (E1)? My brain hurts.

 

[tiny-tim]

Does

gamma*m*c^2 = p1c + p2c

look good?

Yes, but split p2 into x and y components, and use Pythagoras.

(going to bed now … goodnight! :zzz:)​

 

[Redbelly98]

Then V = C - V^2/C? Doesn't make sense to me...

If you solve this quadratic equation in v, you'll get the answer you are supposed to get.

actually, V²/c = 1 - V²/c, but still gives the wrong answer

There is a problem with inconsistent units here

mathman44, i don't know how you solved for E1, but you do need cons. of energy

Hmmm, I'm also puzzled how E1 was solved for. But mathman44's equation does yield the same answer given in the problem statement.

 

[Redbelly98]

... my teacher gave us E1 in class.

In that case, you don't need conservation of energy. Presumably it was already used by your teacher to determine E1.

Just solve the quadratic equation for v you had earlier.

 

[mathman44]

Great, thanks, that works perfect!