# Homework Help: Calculate the velocity of the centre of mass at threshold

1. Jan 1, 2013

### unscientific

1. The problem statement, all variables and given/known data

A high-energy photon γ1 of energy 261 GeV collides with an infra-red photon γ2 of energy 1eV to produce an electron-positron pair via the reaction

γ1 + γ2 → e- + e+

3. The attempt at a solution

Conservation of momentum:

p1 - p2 = (0.5)pe

But since p1 is 1011 times more than p2,

pe ≈ (0.5)(p1) = 6.96 x 10-17 kg m s-1

But when i try to calculate the velocity of the positron/electron using the momentum i get v = c...

Comparing orders of magnitude

pe ≈ 10-17

m(γv) = (10-23) (γ)

This implies that γ ≈ 106....

Am I doing something wrong here?

2. Jan 1, 2013

### Staff: Mentor

First, working in SI units is a bit impractical here - you get large powers of 10 everywhere.
Second: How can you know that the reaction is at threshold? Otherwise p1 - p2 = (0.5)pe is wrong.

You should get something really close to, but below c.

The order of magnitude is correct.

3. Jan 1, 2013

### unscientific

I tried to calculate it but my calculator simply gives c...

4. Jan 1, 2013

### Staff: Mentor

$\frac{v-c}{c} \approx 5 \cdot 10^{-13}$, you would need ~13 digits to see a difference.

5. Jan 1, 2013

### unscientific

Is this question a trick question?

6. Jan 1, 2013

### Staff: Mentor

No, why?
What is the problem statement, by the way? What are you supposed to calculate?

7. Jan 2, 2013

### unscientific

It's a 7 marks question, where i'm supposed to find the velocity of the positron/electron pair at threshold energy.

8. Jan 2, 2013

### Staff: Mentor

Well, you can give the relative deviation from c as result (similar to my post), as the value in m/s is not really interesting.

9. Jan 2, 2013

### unscientific

I see, so this is actually an approximation question?

10. Jan 2, 2013

### Staff: Mentor

I don't understand your question. You can give an exact result, where is the problem?

11. Jan 3, 2013

### unscientific

The problem is that my calculator's only able to calculate up to 8 decimal places.. so it can only give 0.99999999c anything beyond that it automatically registers c.

12. Jan 3, 2013

### Staff: Mentor

You can calculate (v-c)/c with a precision of 8 decimal places then.

13. Jan 4, 2013

### unscientific

Here's what I got; 12 decimal places. When i substituted it into my calculator it gave me a value for gamma as 7*10^5

14. Jan 4, 2013

### Staff: Mentor

$\beta = \sqrt{1-10^{-12}} \approx 1-\frac{1}{2}10^{-12}$
$1-\beta \approx 5 \cdot 10^{-13}$
This allows to determine (1-β) with the same relative precision as γ.