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Data Analysis
Exercises continued
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Exercise 4
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Part 1
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Muons (also called
µ mesons) are unstable particles. A muon decays into
(changes into) an electron, a neutrino and an anti-neutrino. The
decay occurs at random but if we have enough particles their
rate of decay is predictable. An experiment was conducted to
observe the rate of decay of muons. The results are shown below.
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time t
/µs |
number of muons
remaining N |
| 0 |
568 |
| 1 |
373 |
| 2 |
229 |
| 3 |
145 |
| 4 |
_99 |
| 5 |
_62 |
| 6 |
_36 |
| 7 |
_17 |
| 8 |
__6 |
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Theory suggests that the equation
which describes this decay has the form
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N = N0 e-lt
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| where |
l is
a constant called the decay constant (for muons) |
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N0
is the number of muons at a certain time |
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N
is the number of muons remaining t seconds later |
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Taking logs of both sides of the
equation gives us:
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a) Plot a graph of N against t.
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b) i) Use three points on the graph to
show that the equation of the graph has the form predicted by the
theory. In doing this calculate the value of the decay constant.
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ii) Use the equation to calculate the
number of muons remaining at t = 6·5µs
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(Verify that your graph gives about
the same result.)
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Part 2
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A similar experiment was conducted
using muons which were moving at high speed relative to the
experimenter. The speed was close to the speed of light. Einstein’s
theory of relativity predicts that in this situation time
for the muons (which we will call "muon time" tm)
will be different from time as measured by the experimenter (te).
Muons, of course, decay at a rate which depends on their own
time not the experimenter’s time.
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In this second experiment, it is found
that at time t = 6·5µs, as measured by the experimenter’s
clock, the number of muons remaining is 400.
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Einstein predicted
that the relation between the experimenter’s time and the muon’s
time is given by:
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where v = the speed of the muons
relative to the experimenter (in this case 2·9805 × 108
ms-1)
and c = the speed of light in a vacuum (2·9979 × 108 ms-1)
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The above equation is often referred to
as the "time dilation" equation.
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Use your graph to show that these two
experiments give support to Einstein’s time dilation equation.
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© David
Hoult 2008 |
The
Open Door Web Site