# Find relative speed of fast moving partical with dilated time

1. Mar 12, 2014

### akennedy

1. The problem statement, all variables and given/known data
The muon is an unstable elementary particle that decays via a weak-force
interaction process into an electron and two neutrinos. The life time of muons in their rest
frame is 2:197 s  2:197106 s. Nuclear reactions in the upper atmosphere, precipitated by
the impact of highly energetic cosmic rays, generate fast-moving muons about 10 km above sea
level. Some of these particles are detected in labs at about sea level. This is possible because
the life time of muons moving with respect to the Earth's surface is dilated.

(i) Estimate the muons' speed relative to Earth from the fact that they travel 10.0 km between
their point of creation in the atmosphere until seen to decay in ground-based labs.
(ii) The proper distance travelled by the muons considered in part (a) is 10.0 km. Calculate
the length of this distance as it would be measured in the muons' rest frame. Explain
how your result, if interpreted using Galilean relativity, would be at odds with nding
cosmic-ray muons at the Earth's surface.

2. Relevant equations
gamma = 1/SQRT(1-v^2/c^2)
Time = gamma*proper time

3. The attempt at a solution
Honestly I'm clueless. I am really struggling with relativity but here's my current understanding of the problem (i). The muon is travelling at high speeds so it experiences time dilation and is able to travel to the surface of the earth without decaying.
The proper time would be the time as measures on the Earth as it is, for all purposes, at rest right?
I'm having trouble understanding the whole dilation aspect... If the partical experiences time dilation does that mean it's only undergone that time above to travel 10KM? Wouldn't that mean it was travelling faster than c?
Is the 2.197x10-6 the proper time or the dilated time? :|

Sorry I really don't know where to start.

2. Mar 13, 2014

### akennedy

I guess my biggest problem is this. The partical must have only experienced 2.197x10-6 seconds but then it's travelling faster than the speed of light in it's frame. Please someone help me understand this lol

I don't know how to solve for v without having both the proper time and dilated time :|

Last edited: Mar 13, 2014
3. Mar 13, 2014

### akennedy

I tried using the t = gamma*tp formular, but replaced v in gamma with 10000/tp and tried to solve for tp but couldn't get anywhere. Can someone please give me a hint?

4. Mar 13, 2014

### vela

Staff Emeritus
Right.

You have two events: the muon's creation and the muon's decay. The proper time would be the time measured by a clock traveling along with the muon between the two events.

In the muon's rest frame, only 2.197 μs elapses between its creation and decay. What you're forgetting is that in the muon's rest frame, the 10-km distance is length-contracted. When you take the length-contracted distance and divide it by 2.197 μs to calculate the muon's speed, you'll get a result less than $c$.

5. Mar 13, 2014

### akennedy

Right, that makes sense. So the proper time is the given time, and I need to find the dilated time to calculate the velocity?

Am I right in that if I find this dilated time, in the Earth's frame, and use it to divide 10,000 I'd get the velocity?
If so, should I just use the t = tp*gamma formular to solve for t while replacing v with 10000/t? Or am I missing something here.

Thanks a lot for your help.

6. Mar 13, 2014

### akennedy

I ended up with 299350496 metres per second. That sounds about right, at least it isn't impossible lol. Thanks a lot buddy.