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Data Analysis Exercises
  Ex 2   Ex 3   Ex 4
   
1. In an experiment, two variable quantities, x and y were measured.
  The relation between x and y is thought to be:
 
   
  The measured values were
 
/m  ±0.01m y  /s ±0.05s
0.10 0.64
0.20 0.90
0.40 1.27
0.80 1.80
1.00 1.90
   
a) Plot a graph which could be used to verify the relation between x and y.
b) From the graph find the value of the constant, a.
c) Add to the graph error bars for the first and last results.
   
2. In an experiment, two variable quantities, x and y were measured.
  The relation between x and y is thought to be:
 
   
  The measured values were
 
/cm  ±0.02cm y  /g ±0.25g
0.67 2.25
0.80 4.00
0.90 5.52
0.95 6.60
   
a) Plot a graph which could be used to verify the relation between x and y.
b) Add error bars to the points on the graph.
c) Use the graph to estimate the maximum and minimum values of the constant, b.
  Give your final answer for b in the usual form:
  b = value ±indeterminacy
d) What physical quantity might b represent?
   
3. A point source of light was placed at different distances, r from a photo-electric cell.
  The current, I generated by the cell was measured.
  It is thought that the relation between I and r is of the form
 
  where k and n are constants.
  Now, (as every ten year old child knows), taking logs of both sides of this equation gives us
 
   
  The measured values were
 
/cm  ±0.5cm I  /mA ±2mA
15 108
20 61
25 39
30 27
35 24
40 15
45 12
   
a) Plot a graph which could be used to verify the relation between I and r.
b) Use the graph to find the value of the constant n
c) Find the value of the constant k
d) A significant error was made in one of the measurements.
  Suggest a possible corrected value for this result.
e) Add error bars to the points representing the first and last results.
   
4. Part 1
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.
   
 
/μs number of muons
remaining N
0 568
1 373
2 229
3 145
4 99
5 62
6 36
7 17
8 6
   
  Theory suggests that the equation which describes this decay has the form
 
 
where   No is the number of muons present at a given time
    N is the number of muons remaining t seconds later
    λ is a constant called the decay constant (for the muons) 
   
  Taking logs of both sides of the equation gives us
 
   
a) Plot a graph of N against t.
b) Use three points on the graph to show that the equation of the graph has the form predicted by the theory and, in doing this, calculate the value of the decay constant.
c) Use the equation to calculate the number of muons remaining at t = 6·5µs
   
  Part 2
  A similar experiment was conducted using muons which were moving at high speed relative to the experimenter.
  The speed was very 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 (see Time Dilation)
  Muons, of course, decay at a rate which depends on their own time not the experimenter’s time.
  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.
   
  Einstein predicted that the relation between the experimenter’s time and the muon’s time is given by
 
 
where   v (the speed of the muons relative to the experimenter) = 2.980108ms-1
    c (the speed of light) = 2.997108ms-1
   
  Use your graph to show that these two experiments give support to Einstein’s time dilation equation.
 
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