1) What is a µ Mesons and where and how are µ Mesons produced ?

	µ Mesons  are produced during nuclear fusion inside the sun
	µ Mesons arrive on Earth as part of the cosmic 'wind'
	µ Mesons decay to form either a electron or a positron together with a neutrino and an anti-neutrino

2) Here is the data for µ Meson decay  as measured in a laboratory rest frame:

    (i) Calculate the decay constant  for muons at rest.

Start with the radioactive decay equation and rearrange it as follows to give an equation for .

Once the equation has been rearranged, substitute in values for n0, nt and t, by choosing some values from the table. If you use the first pair of values you will get a value of  of 0.45×10-6 s-1

    (ii) Using this constant, calculate the time that should have elapsed in the muon's rest frame for the count rate to have fallen from 563 hr^-1 to 400 hr^-1

Using the radioactive decay equation , rearrange it to give an equation for t.

Once the equation has been rearranged substitute in 563 for n0, 400 for nt and your value of to find t. t should be approximately 0.76×10-6 s.

3) Show that mesons take about 6.5×10^-6 s to reach sea level from an altitude of 2000m. Show also that the muon count rate should have fallen to about 25 hr^-1 by the time the mesons had reached sea level.

	First calculate the time for the muons, which are assumed to be traveling at the speed of light (3×108 ms-1)
	Next use the radioactive decay equation to calculate the count rate after 6.5×10-6 s, take n0 as 563, and substitute your value of in to find nt. If you work in ×10-6 s it will make the sums easier.

4) Deduce that the time dilation factor (T₀/t) is approximately ¹/₉. Hence establish that the muons travel at a speed of roughly 0.994c.

	Your time calculated in 2(ii) is the time from the muons frame of reference T0, the time calculated in 3 is the time from an observers frame of reference, therefore T0/t should be approximately 0.114 or 1/9.
	One you have T0/t you can substitute this into the time-dilation formula to find v/c

now just substitute in the values of T₀ and t, this will give you v/c.

5) Explain fully the statement that 'If we take two groups of unstable particles of the same type, giving a group speed of v and letting the other remain at rest with respect to our frame of reference., the radioactive clock represented by the decay in the moving group will run slow compared to that of the stationary group'.

    This question asks you to explain the process of time dilation, and how it relates to radioactive decay, the key points to mention are

	The stationary particles do not experience any time dilation, they can be seen to experience T0
	The moving particles experience time passing at a slower rate
	In this situation number of radioactive decays is taken to be equivalent to the time
	Therefore the stationary particles will have decayed more than the moving particles

6) From this article calculate the %age of the muons which are measured by the Earthbound observer to have reached sea level. Calculate also the %age of muons that would reach the observer if there was a 'Universal Newtonian Time'.

    In the first part of the question the % of muons reaching sea level is given by (Muon Count rate at sea level/Muon Count Rate at 6300ft)×100%.

    The number of muons at sea level is 412hr^-1

    The Muon count rate at 6300ft is 568 hr^-1

   The answer is 72.53%

    In the second part of the question you are asked to calculate the percentage of muons reaching sea level if time dilation does not exist, for this one the only difference is the Muon count rate at sea level is 27 hr^-1

  The Answer is 4.75%

7) If Muons moving at speed v travel distance H in time ?t as measured by an observer on Earth, postulate that from their frame of reference they travel a distance H' in time ?t'.

    i) Show that this leads to the equation

    Start with the time dilation equation

Given that

Therefore

For T₀

For t

Therefore

Finally remove 'v'

ii) Interpret this equation physically

    This means you must draw your conclusions on what the equation means

	As Speed increases, apparent distance traveled decreases
	At v=c H'=0
	At v=0 H=H'

iii) Calculate the distance H' that the muons in Rossi and Halls' experiment cover as measured in their own rest frame

    All you need to do is substitute the following data into the equation you derived in part i

H=2000m v=0.994c

Correct answer is 217.68m

8) Equation * is the Lorentz-Fitzgerald contraction of length.

    i) Do Objects appear longer in a frame at rest or a moving frame

    Objects appear longer in a stationary frame of rest

    ii) Which is correct

where L0 is the 'rest length' or 'proper length'
The correct equation is

iii) Plot a graph of L/L₀ against v/c and comment

 The graph should look something like this

9) An unstable particle has a half life of 2 µ s and travels at speeds within 0.2% of light speed

    i) Calculate the distance that it appears to cover before decay in its own rest frame

v=.998c

v=2.994×10⁸ ms^-1

s=2.994×10⁸ ×2×10^-6

s=598.8m

    ii) Calculate the distance that it would appear to travel relative to a fixed observer on Earth. Would a significant number of these particles reach the ground if they were created at an altitude of 10km?

This time we must use relativity , and the formula we found in 7(i)

in this case we need to find L₀

substitute in L and v/c, this will give L₀ = 9472.60 m

This means that quite a few particles will reach ground level from an altitude of 10km