Finding the total energy of a Pi meson

[AsadaShino92]

Homework Statement

A K meson (an elementary particle with approximately 500 Mev rest mass) traveling through the laboratory breaks up into two Pi mesons (elementary particles with 140 Mev rest energies). One of the Pi mesons is left at rest. What is the total energy of the remaining Pi meson?

Homework Equations

E=mc^2
KE=(γ-1)mc^2

The Attempt at a Solution

My first step was trying to use energy conservation E(initial)=E(final)
The K meson is traveling so the total E at the start is E(rest mass of K)+KE(K meson)
E(initial)=500 Mev+(γ-1)mc^2

In the end, the K meson breaks into 2 Pi mesons, each with 140 Mev rest mass. One stops and the other keeps on moving.
So the total E at the end is E(rest mass Pi)+E(rest mass 2nd Pi)+(γ-1)mc^2
E(final)=140 Mev+140Mev+(γ-1)mc^2

E(initial)=E(final)
500 Mev+(γ-1)mc^2=140 Mev+140Mev+(γ-1)mc^2

But now I don't know what to do next. I don't have a value for gamma.

 

[PeroK]

Homework Statement

A K meson (an elementary particle with approximately 500 Mev rest mass) traveling through the laboratory breaks up into two Pi mesons (elementary particles with 140 Mev rest energies). One of the Pi mesons is left at rest. What is the total energy of the remaining Pi meson?

Homework Equations

E=mc^2
KE=(γ-1)mc^2

The Attempt at a Solution

My first step was trying to use energy conservation E(initial)=E(final)
The K meson is traveling so the total E at the start is E(rest mass of K)+KE(K meson)
E(initial)=500 Mev+(γ-1)mc^2

In the end, the K meson breaks into 2 Pi mesons, each with 140 Mev rest mass. One stops and the other keeps on moving.
So the total E at the end is E(rest mass Pi)+E(rest mass 2nd Pi)+(γ-1)mc^2
E(final)=140 Mev+140Mev+(γ-1)mc^2

E(initial)=E(final)
500 Mev+(γ-1)mc^2=140 Mev+140Mev+(γ-1)mc^2

But now I don't know what to do next. I don't have a value for gamma.

What about momentum?

 

[AsadaShino92]

What about momentum?

Then I would use p=γmv but I don't think I have the value for those. I just know that momentum would also be conserved.

 

[PeroK]

Then I would use p=γmv but I don't think I have the value for those. I just know that momentum would also be conserved.

In general it's messy to use $\gamma$ in this sort of problem. Can you think of a way to tackle the problem without it?

 

[AsadaShino92]

In general it's messy to use $\gamma$ in this sort of problem. Can you think of a way to tackle the problem without it?

P=E/C ?

 

[PeroK]

P=E/C ?

That's for a massless particle. Do you know:

$E^2 = p^2c^2 + m^2c^4$?

 

[AsadaShino92]

That's for a massless particle. Do you know:

$E^2 = p^2c^2 + m^2c^4$?

So am I using this equation to solve for momentum?

 

[PeroK]

So am I using this equation to solve for momentum?

The general approach to these particle problems is:

1) Conservation of energy
2) Conservation of momentum.
3) For each particle: $E^2 = p^2c^2 + m^2c^4$

Hint: stick with $M$ for the initial K meson mass and $m$ for the pi meson mass and leave the numbers out until the end.

The notation is up to you, but I would use $E_0, p_0$ for the Energy & Momentum of the initial K meson; $E_1, p_1 (p_1= 0)$ for the pi meson that ends up at rest and $E_2, p_2$ for the second pi meson.

You're trying to solve for $E_2$ of course.

 

[AsadaShino92]

The general approach to these particle problems is:

1) Conservation of energy
2) Conservation of momentum.
3) For each particle: $E^2 = p^2c^2 + m^2c^4$

Hint: stick with $M$ for the initial K meson mass and $m$ for the pi meson mass and leave the numbers out until the end.

The notation is up to you, but I would use $E_0, p_0$ for the Energy & Momentum of the initial K meson; $E_1, p_1 (p_1= 0)$ for the pi meson that ends up at rest and $E_2, p_2$ for the second pi meson.

You're trying to solve for $E_2$ of course.

Using the notation, I have the conservation of energy written as E₀=E₁+E₂
Then using the energy equation for each particle

For the K meson
E₀²=P₀²C²+M²C⁴

For the Pi stopped
E₁²=m²C⁴ since P₁=0

For Pi moving
E₂²=P₂²C²+m²C⁴

Momentum conservation
P₀=P₁+P₂
Since P₁=0
P₀=P₂

Do I just plug all this into E₂=E₀-E₁?

 

[PeroK]

Do I just plug all this into E₂=E₀-E₁?

Why not? You may want to square that equation first.

 

[AsadaShino92]

Why not? You may want to square that equation first.

You mean instead I should use E₂²=E₀²-E₁²?

 

[PeroK]

You mean instead I should use E₂²=E₀²-E₁²?

$(E_0 - E_1)^2 \ne E_0^2 - E_1^2$

 

[AsadaShino92]

$(E_0 - E_1)^2 \ne E_0^2 - E_1^2$

My mistake. Then squaring both sides should be E₂²=(E₀-E₁)²

 

[PeroK]

My mistake. Then squaring both sides should be E₂²=(E₀-E₁)²

$E_2^2 = (E_0 - E_1)^2 = \dots$?

 

[AsadaShino92]

$E_2^2 = (E_0 - E_1)^2 = \dots$?

(E₀-E₁)²=E₀²-2E₀E₁+E₁²

 

[PeroK]

(E₀-E₁)²=E₀²-2E₀E₁+E₁²

Okay, but you may need to do more than one line of algebra at a time. I'm going off-line now, but maybe someone else can provide further help if you need it.