# Calculating Muon Half-Life: Length Contraction vs Time Dilation

• mananvpanchal
In summary, the two different equations give two different results because they are calculating two different things.
mananvpanchal

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html

In this link, with non-relativistic method, half life is calculated as 21.8.

But with relativistic method half life is calculated as 4.36.

We calculate half life considering time dilation for muon in Earth frame.
And, we calculate half life considering length contraction for muon in muon frame.
We get same value using both method as 4.36.

So, the question is:
Would length contraction and time dilation both not exist for muon?
Would we not include both factor in equation to calculate half life?
Why are we using length contraction and time dilation one by one to calculate half life?

mananvpanchal said:
In this link, with non-relativistic method, half life is calculated as 21.8.

But with relativistic method half life is calculated as 4.36.
Those are not half lives, but the travel time expressed in terms of half life.
We calculate half life considering time dilation for muon in Earth frame.
And, we calculate half life considering length contraction for muon in muon frame.
We get same value using both method as 4.36.
Both observers agree that 4.36 half lives will occur during the muon's travel to the earth.
So, the question is:
Would length contraction and time dilation both not exist for muon?
Would we not include both factor in equation to calculate half life?
Why are we using length contraction and time dilation one by one to calculate half life?
The muon travels from some distance above the Earth to the Earth's surface. In the muon's frame, it has a certain half-life. Viewed from the earth, time dilation will apply to the moving muon's half-life; length contraction is not relevant since the distance is in the Earth frame. Viewed from the muon's frame, it's the distance traveled that will be contracted; time dilation is not relevant since the muon is at rest in its own frame. Those end up being two different yet equivalent descriptions of what happens; each description leads to the same observable result.

## 1. What is the concept of length contraction and time dilation?

Length contraction is the phenomenon where an object appears shorter in the direction of its motion relative to an observer. Time dilation is the effect where time appears to pass slower for an object that is moving at high speeds relative to an observer.

## 2. How do length contraction and time dilation impact the calculation of muon half-life?

Length contraction and time dilation affect the measurement of time for a muon, which travels at high speeds in Earth's atmosphere. This results in a longer half-life for the muon, as it experiences less time compared to an observer on Earth.

## 3. What is the formula for calculating muon half-life in the presence of length contraction and time dilation?

The formula for calculating muon half-life is t = t0 / √(1-v2/c2), where t is the observed half-life, t0 is the half-life at rest, v is the velocity of the muon, and c is the speed of light.

## 4. How do we measure the velocity of muons in order to use the formula for calculating half-life?

The velocity of muons can be measured using a variety of methods, including detectors and imaging techniques. These measurements are then used in the formula to calculate the half-life of the muon.

## 5. Are there any other factors that can affect the calculation of muon half-life?

Yes, in addition to length contraction and time dilation, there are other factors that can affect the calculation of muon half-life, such as the presence of magnetic fields and interactions with other particles in the atmosphere. These factors must also be taken into account in order to accurately calculate the muon half-life.

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