# Special relativity - kinematics

• Aleolomorfo
In summary, the conversation discussed a process where a photon hits a proton at rest in the laboratory frame, resulting in the creation of a neutron and a positively charged pion. The threshold energy of the photon was found using the invariance of total momentum squared between the lab and the center of mass frame. The mean lifetime of the pion in the lab frame was then calculated, taking into account the mean lifetime at rest and the pion's velocity in the lab frame. The assumption that the particles move with the same velocity in the lab frame was confirmed, and the conservation of energy was used to find the pion's velocity and mean lifetime.
Aleolomorfo

## Homework Statement

A photon hits a proton at rest in the laboratory frame and there is the process:
$$\gamma + p \rightarrow n+\pi^+$$
The mass of the pion is ##m_\pi## and assuming that the masses of the proton and the neutron are the same (##m##):
1. Finding the threshold energy of the foton;
2. At the threshold energy, finding the mean lifetime of the pion in the laboratory frame knowing that the mean lifetime at rest is ##\tau_0##

## The Attempt at a Solution

For the first point I have used the invariance of the total momentum squared bewtween the lab frame before the collision and the CM frame after the collision (particle at rest) and my result is:
$$E=m_\pi + \frac{m_\pi^2}{2m}$$
I think this is correct.
For the second part I need a confirmation about my reasoning. I need to calculate the pion's velocity in the lab and then ##\tau=\gamma \tau_0##. Since at the minimum energy the particles are at rest in the CM frame, is it correct to state that in the lab frame they move with the same velocity along the same direction(both to the right or equivalently to the left)? If this assumption is correct I can use the conservation of the energy ##E+m=m\gamma+m_{\pi}\gamma##. From this relation I can find ##\gamma## and so ##\tau##. Is it correct?

It all looks correct to me.

Aleolomorfo
TSny said:
It all looks correct to me.
Perfect, thank you!

## 1. What is special relativity and how does it differ from general relativity?

Special relativity is a theory developed by Albert Einstein that describes the relationships between space and time for objects moving at constant speeds. It differs from general relativity in that it does not take into account gravitational effects, while general relativity covers both gravity and motion.

## 2. How does time dilation work in special relativity?

Time dilation is the phenomenon where time appears to move slower for objects moving at high speeds compared to stationary objects. This is due to the fact that as an object's speed increases, its perception of time slows down, as described by the equation t' = t/(sqrt(1-(v^2/c^2))), where t' is the observed time, t is the proper time, v is the speed of the object, and c is the speed of light.

## 3. What is length contraction and how is it related to special relativity?

Length contraction is the phenomenon where objects appear to be shorter in the direction of motion when moving at high speeds compared to when they are stationary. This is due to the fact that as an object's speed increases, its perception of distance decreases, as described by the equation l' = l*sqrt(1-(v^2/c^2)), where l' is the observed length, l is the proper length, v is the speed of the object, and c is the speed of light.

## 4. Can special relativity be applied to objects moving at speeds close to the speed of light?

Yes, special relativity can be applied to objects moving at any speed, including speeds close to the speed of light. This is because the theory is based on the principle of relativity, which states that the laws of physics are the same for all observers in uniform motion.

## 5. How does the concept of simultaneity change in special relativity?

In special relativity, the concept of simultaneity is relative and depends on the observer's frame of reference. This means that two events that are simultaneous for one observer may not be simultaneous for another observer. This is due to the fact that the speed of light is always constant, regardless of the observer's frame of reference, and this affects the perception of time and space.

Replies
4
Views
865
Replies
2
Views
1K
Replies
6
Views
4K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
19
Views
1K
Replies
3
Views
1K
Replies
6
Views
2K