# Calculate the velocity of the centre of mass at threshold

• unscientific
In summary: 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##\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.
unscientific

## Homework Statement

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

γ1 + γ2 → e- + e+

## 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?

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.

But when i try to calculate the velocity of the positron/electron using the momentum i get v = c...
You should get something really close to, but below c.

This implies that γ ≈ 106...
The order of magnitude is correct.

mfb said:
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.

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

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

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

Is this question a trick question?

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

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

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

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.

mfb said:
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.

I see, so this is actually an approximation question?

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

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

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.

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

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

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

##\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.

## 1. What is the formula for calculating the velocity of the centre of mass at threshold?

The formula for calculating the velocity of the centre of mass at threshold is Vcm = √(2gh), where Vcm is the velocity of the centre of mass, g is the gravitational acceleration, and h is the height of the object at threshold.

## 2. How is the centre of mass related to an object's velocity at threshold?

The centre of mass is the point where an object's mass is evenly distributed, and it is directly related to the object's velocity at threshold. As the centre of mass moves, the object's velocity also changes.

## 3. Can the velocity of the centre of mass at threshold be negative?

Yes, the velocity of the centre of mass at threshold can be negative. This indicates that the centre of mass is moving in the opposite direction to the initial motion of the object.

## 4. How does the height of an object affect the velocity of its centre of mass at threshold?

The height of an object directly affects the velocity of its centre of mass at threshold. The higher the object is at threshold, the greater the velocity of its centre of mass will be.

## 5. What is the significance of calculating the velocity of the centre of mass at threshold?

Calculating the velocity of the centre of mass at threshold is significant because it helps us understand the motion and stability of objects. It also allows us to make predictions about the behaviour of objects at different heights and speeds.

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