Question

Suppose a particle is created by cosmic radiation striking the atmosphere. It moves by you at 0.340c, and it lives 47.7533 microseconds when at rest relative to an observer. How long does it live (in microseconds) as you observe it?

Answer 1

The lifetime of a particle moving at 0.340c as observed by a stationary observer is calculated using the time dilation formula from the theory of special relativity, which results in an approximate lifetime of 50.7492 microseconds.

Step-by-step explanation:

The question involves the concept of time dilation in the theory of special relativity, which was developed by Albert Einstein. Time dilation refers to the phenomenon where time runs slower for an object in motion compared to an object at rest, observable especially at velocities approaching the speed of light (denoted as 'c'). The lifetime of a moving particle appears longer to a stationary observer because of this time dilation.

To calculate how long the particle lives as you observe it, you can use the time dilation formula:

Lifetime as observed = Lifetime at rest / sqrt(1 - v^2/c^2)

Filling in the values:

• Velocity, v = 0.340c
• Lifetime at rest = 47.7533 microseconds
• Speed of light, c = 1 (since 'v' is in units of 'c')

We get:

Lifetime as observed = 47.7533 / sqrt(1 - 0.340^2)

Calculating the square root gives us:

sqrt(1 - 0.340^2) = sqrt(0.8844) ≈ 0.9409

So, the lifetime as observed ≈ 47.7533 / 0.9409 μs

Therefore, the particle's observed lifetime is approximately 50.7492 microseconds.