How Does an Unstable Particle Breakup Affect the Speed of Its Fragments?

[frankR]

An unstable particle at rest breaks up into two fragments of unequal mass. The rest mass of the lighter particle is 2.5 x 10^-28 kg, and that of the heavier fragment is 1.67 x 10^-27 kg. If the lighter fragment has a speed of 0.983c after the breakup, what is the speed of the heavier fragment?


What is the idea here? Where does the energy come from, from an external source, or from the mass in the particles? (note: lower case m corresponds to the mass of the light particle and upper case M corresponds to the mass of the heavy particle)

v₁: speed of the lighter particle
v₂: speed of the heavier particle

We know that the total relativistic energy is:

E = KE + m_oc²

KE = 1/2mv₁², m is the relativistic mass

So if relativistic energy of the light particle is conserved we get this equation.

mc² = 1/2mv² + m_oc²

We can also conserve the relativistic mass of both particles:

(m + M)c² = 1/2mv₁² + 1/2Mv₂² + (m_o + M_o)c²

Is this the correct way to setup the problem?

I tried solving for v₂ without any luck. So I hope there is an easier way--the correct way.

Thanks

 

[quantumdude]

Originally posted by frankR
What is the idea here?

The idea is to apply conservation of energy and momentum to a relativistic problem.

Where does the energy come from, from an external source, or from the mass in the particles?

It comes from the mass of the inital body. When you are done solving the problem, you can verify that the combined masses of the fragments is less than the mass of the whole. Rest mass energy was converted to kinetic energy.

(note: lower case m corresponds to the mass of the light particle and upper case M corresponds to the mass of the heavy particle)

That's a bad choice, in my opinion. I would use capital M for the mass of the body before disintegration, and m_L, m_H for the masses of the light and heavy fragments, respectively. Descriptive variable names can help you track which quantity is which throughout the problem.

v₁: speed of the lighter particle
v₂: speed of the heavier particle

We know that the total relativistic energy is:

E = KE + m_oc²

OK

KE = 1/2mv₁², m is the relativistic mass

No, that's nonrelativistic KE. Look up relativistic KE in your book. It is:

KE=(γ-1)mc²

and if you add that to the rest mass energy to get the total energy of a free particle, you get:

E=mc²+KE=mc²+(γ-1)mc²
E=γmc²

This should all be in your book.

So if relativistic energy of the light particle is conserved we get this equation.

mc² = 1/2mv² + m_oc²

No, even if you were right about KE you wouldn't get that. The energy of the light particle by itself is not conserved.

We can also conserve the relativistic mass of both particles:

(m + M)c² = 1/2mv₁² + 1/2Mv₂² + (m_o + M_o)c²

No, on the left you have the total mass equal to (m+M), which is not true.

I am going to use the notation I suggested.

The total energy prior to disintegration is:

E_i=Mc²

The total energy after disintegration is:

E_f=γ_Lm_Lc²+γ_Hm_Hc²

Let E_i=E_f, and you have the equation for conservation of energy. Since you have two unknowns (M and v_H), you need another equation. Luckily, conservation of momentum saves the day.

Total momentum before disintegration:

p_i=0

Total momentum after disintegration:

p_f=γ_Lm_Lv_L+γ_Hm_Hv_H

Let p_i=p_f, and you have the equation for conservation of momentum.

edit: fixed variable subscripts

 

[frankR]

I got v_H = .3265*c

However the correct answer should be .285*c

I founs these equations:

From: p_L = p_H I got:

V_H = (Y_L/Y_H)*(m_Lo/m_Ho)*V_L

From E_i = E_f:

I found:

Y_H = (m_Lo + m_Ho - Y_L*m_Lo)/m_Ho

I then substited Y_H in V_H = (Y_L/Y_H)*(m_Lo/m_Ho)*V_L and calculated my answer.

What went wrong?

Thanks.

 

[quantumdude]

Originally posted by frankR
From E_i = E_f:

I found:

Y_H = (m_Lo + m_Ho - Y_L*m_Lo)/m_Ho

How did you eliminate the mass M of the initial particle?

 

[frankR]

Originally posted by Tom
How did you eliminate the mass M of the initial particle?

Uhhhhh... By making it go to zero.

Nevertheless you can solve it strickly with ph=pl.

 

[quantumdude]

Originally posted by frankR
Nevertheless you can solve it strickly with ph=pl.

Yes, that's true--I didn't notice that at first.

So you got it then?

 

[frankR]

Yeah, I fiquered it out right before I turned it in. It was a Newton's Third Law problem with a Special Relativity twist.