# Relativity: Solving Pion Homework Problem

In summary, the question asks for the speed of a positive pion (π+) that must travel down a 1.00 km long tube without decaying before it reaches the end. The particle has an average lifetime of 2.60×10−8s in its rest frame and a rest energy of 139.6 MeV. Using the equations for time dilation and total energy, we can calculate the speed to be (1-Δ)c, where Δ is a small value representing the difference between the speed and the speed of light. For the second part of the question, we need to use the correct relativistic momentum equation and include the factor of gamma to find the total energy.

## Homework Statement

After being produced in a collision between elementary particles, a positive pion (π+) must travel down a 1.00 km -long tube to reach an experimental area. A π+ particle has an average lifetime (measured in its rest frame) of 2.60×10−8s; the π+ we are considering has this lifetime.
How fast must the π+ travel if it is not to decay before it reaches the end of the tube? (Since u will be very close to c, write u=(1−Δ)c and give your answer in terms of Δ rather than u.)
The π+ has a rest energy of 139.6 MeV. What is the total energy of the π+ at the speed calculated in part A?

## Homework Equations

Δ t = Δt0 / sqrt(1-u^2/c^2)

E^2 = (mc^2)^2 + (pc)^2

## The Attempt at a Solution

I got the correct answer for speed, the first part of the question. It's the second part I can't get to work. I used the total energy equation and my speed, which worked out to give me E = 197.4 MeV but this wasn't right. I'm not sure where I'm going wrong?

I used the total energy equation and my speed, which worked out to give me E = 197.4 MeV but this wasn't right. I'm not sure where I'm going wrong?
We will not be able to tell you this unless you actually show us what you did, not just try to describe it in words.

Orodruin said:
We will not be able to tell you this unless you actually show us what you did, not just try to describe it in words.

E^2 = (mc^2)^2 + (pc)^2
I used mc^2 = 139.6 MeV
I put p = mv so the pc = mvc but m = 139.6/c^2 and v = (1-Δ)c = (1-(3.04*10-5))c so pc = 139.6 MeV
So then E = sqrt(139.6^2 + 139.6^2) = 197.4 MeV

I put p = mv
This is not the relativistic momentum. This relation is only valid at non-relativistic speeds.

Orodruin said:
This is not the relativistic momentum. This relation is only valid at non-relativistic speeds.

Yes, that makes sense. I forgot to include gamma. I got the correct answer now, thanks.

## 1. What is the theory of relativity?

The theory of relativity is a scientific theory developed by Albert Einstein in the early 20th century. It consists of two main components: the special theory of relativity and the general theory of relativity. The theory explains the relationship between space and time, and how they are affected by the presence of mass and energy.

## 2. How does relativity apply to the Pion homework problem?

The Pion homework problem involves calculating the time dilation effect on a Pion particle as it travels close to the speed of light. This effect is explained by the special theory of relativity, which states that time and space are relative to the observer's frame of reference.

## 3. What is the equation for calculating time dilation in relativity?

The equation for time dilation in relativity is t' = t/√(1-v²/c²), where t' is the time measured by the observer, t is the time measured by the moving object, v is the relative velocity between the observer and the object, and c is the speed of light.

## 4. How does the theory of relativity impact our understanding of the universe?

The theory of relativity has greatly impacted our understanding of the universe by providing a more accurate and comprehensive explanation of the relationship between space, time, and gravity. It has also led to the development of important scientific concepts, such as black holes and the expanding universe.

## 5. Are there any real-world applications of relativity?

Yes, there are many real-world applications of relativity. For example, the GPS system relies on the principles of relativity to accurately calculate the position and time on Earth. Relativity is also crucial in fields such as astronomy, particle physics, and cosmology.

Replies
3
Views
990
Replies
13
Views
3K
Replies
2
Views
3K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
5
Views
5K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
4
Views
3K
Replies
34
Views
3K