How to calculate the minimum kinetic energy (special relativity)?

[Efeguleroglu]

Homework Statement

A moving electron collides to an stable electron and in conclusion additionaly 1 electron and 1 positron occur. At the end if all the 4 particles have the same velocity, we can say the kinetic energy required for this process is minimum. Show that KE(min)=6mc^2 (m is the rest mass of an electron)

Relevant Equations

KE=γmc^2-mc^2

Sadly, that's what all I could do.

 

[vela]

I find it's better to work with energy and momentum, rather than writing things in terms of velocities.

You've used conservation of energy. What relationship do you get if you use conservation of momentum?

 

[Efeguleroglu]

I still can't get the answer. I thought I didn't understand the question and calculated for both inital kinetic energy and final kinetic energy ( I know the change in kinetic energy is -mc^2 from the conservation of energy) but couldn't get the answer. There shouldn't be any calculation mistakes because I used a calculator.

 

[PeroK]

That's a lot of work!

1) What can you say about the total momentum after the collision?

2) What can you say about the momentum of each particle after the collision?

 

[Efeguleroglu]

The momentum of the particles are equal. And that's the momentum which must be conserved.

 

[PeroK]

The momentum of the particles are equal. And that's the momentum which must be conserved.
View attachment 245271

You've gone wrong, I'm sorry to say. Best to start again. Here are some general hints:

a) Find the total energy of the moving electron. You can easily get the KE from this. Splitting the energy into $T + mc^2$ too soon just complicates the equations.

b) You have 3 key equations:

Conservation of total energy
Conservation of total momentum
The energy-momentum-mass equation for each particle ($E^2 = p^2c^2 + m^2c^4$).

Concentrate on those.

 

[Efeguleroglu]

There must be a problem in this question. It's ridiculous but V(i) is less than it should be thus I found V(f) is imaginary. I didn't spot any mistake. Sorry for bothering you with this :(

 

[PeroK]

Again, it's difficult to see where you are going wrong. This should only be 3-4 lines of algebra using the energy and momentum equations. It's always messy when you try to solve for $v$.

I would start as follows:

Let $E_0, p_0$ be the energy and momentum of the moving electron; and, $E_1, p_1$ be the energy and momentum of each of the particles after the collision.

Conservation of energy gives:

$E_0 + mc^2 = 4E_1 \ \ \ (1)$

Conservation of momentum gives:

$p_0 = 4p_1 \ \ \ (2)$

And, we have:

$E_0^2 = p_0^2c^2 + m^2c^4 \ \ \ (3a)$
$E_1^2 = p_1^2c^2 + m^2c^4 \ \ \ (3b)$

The next obvious step is to square equation (1), and then do some substitutions. It should come out in a couple of steps.

 

[vela]

You made an error pretty much from the start. The fourth line is wrong because you did something akin to $(a+b)^2 = a^2 + b^2$.

As I suggested yesterday and as @PeroK has again suggested, work in terms of energy and momentum.

 

[Efeguleroglu]

You made an error pretty much from the start. The fourth line is wrong because you did something akin to $(a+b)^2 = a^2 + b^2$.

As I suggested yesterday and as @PeroK has again suggested, work in terms of energy and momentum.

Ahh yes, I saw it. I applied the momentum-energy equation from particles to the set of particles wrongly. Thanks, you helped me to spot that.

 

[Efeguleroglu]

Again, it's difficult to see where you are going wrong. This should only be 3-4 lines of algebra using the energy and momentum equations. It's always messy when you try to solve for $v$.

I would start as follows:

Let $E_0, p_0$ be the energy and momentum of the moving electron; and, $E_1, p_1$ be the energy and momentum of each of the particles after the collision.

Conservation of energy gives:

$E_0 + mc^2 = 4E_1 \ \ \ (1)$

Conservation of momentum gives:

$p_0 = 4p_1 \ \ \ (2)$

And, we have:

$E_0^2 = p_0^2c^2 + m^2c^4 \ \ \ (3a)$
$E_1^2 = p_1^2c^2 + m^2c^4 \ \ \ (3b)$

The next obvious step is to square equation (1), and then do some substitutions. It should come out in a couple of steps.

Thank you! It appeared quickly.

 

[PeroK]

Thank you! It appeared quickly.
View attachment 245278

Hey, that was easy!