Relativistic Situations (Edexcel International A Level Physics)
Revision Note
Relativistic Situations
Accelerated particles often reach speeds that are very close to the speed of light
At such high velocities and energies, relativistic effects begin to become important
These are effects such as:
Time dilation
Length contraction
Time Dilation
Clocks run slower for moving particles
This means that unstable particles, with a short lifetime, actually survive for much longer in a laboratory if they are moving very quickly
This is useful because they will leave longer tracks in particle detectors (making detection easier)
For example, muons created high up in the atmosphere, have a lifetime of about 2 μs
The time required to travel to sea-level is too great to survive the journey
However, they are detected at sea-level in large numbers
This is because they are travelling at relativistic speeds (e.g. 0.98c) so time dilation means their lifetime is dilated to times much longer than 2 μs
Because muons travel so quickly, time dilation stretches muon lifetime to much longer than when they are at rest
Length Contraction
Moving rulers are shorter than stationary rulers
This means that particles moving at very high velocities travel much further through detectors than expected
Unstable particles with very short lifetimes would not travel for appreciable distances without relativistic effects like length contraction
For example, if exotic particles produced in particle accelerators decayed within the particle chamber before escaping it, none would be detectable
In fact, many types of exotic particles are detected
This is evidence of length contraction
Worked Example
Muons, which normally have a lifetime of 2.2 × 10–6 s, are created in the upper atmosphere at a height of about 10 km above sea level.
a) Calculate the distance a muon would travel towards the ground if it was moving at 0.99 c.
b) Comment on the relativistic effects necessary if muons are to be detected at sea level.
Answer:
Part (a)
Step 1: Write the known quantities
Muon lifetime, t = 2.2 × 10–6 s
Speed of light, c = 3 × 108 m s–1
Speed of muons, v = 0.99 c = 0.99 × (3 × 108) = 2.97 × 108 m s–1
Step 2: Calculate distance travelled
Speed v = distance d ÷ time t
Therefore, the distance travelled by muons travelling at 0.99 c is given by:
d = vt = (2.97 × 108) × (2.2 × 10–6) = 653.4 m
Part (b)
Step 1: Compare the distance calculated to the distance required
The distance a muon travels with a lifetime of 2.2 × 10–6 s is only 653.4 m
This is much less than the 10 km required to sea level
Step 2: Conclude that relativistic effects must be at play
Therefore, time dilation must be allowing the muons to last much longer than 2.2 × 10–6 s
This is because they are detected in large numbers at sea level
Examiner Tips and Tricks
For your exam, you are only required to understand the situations in which a relativistic increase particle lifetime would be significant. As seen in the worked example, this is a combination of time dilation and length contraction. This is when, as we have seen, particles are moving very close to the speed of light. This is normally at velocities greater than 90% the speed of light.
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