# Relativistic Situations(Edexcel International A Level Physics)

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Katie M

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Physics

## 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.

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:

dvt = (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

#### Exam Tip

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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