An unstable high-energy particle enters a detector and leaves a track 1.05 mm long before it decays. Its speed relative to the detector was 0.992c. What is its proper lifetime? That is, how long would it have lasted before decay had it been at rest with respect to the detector?

To find the proper lifetime of the unstable high-energy particle, we can use the concepts of special relativity, particularly time dilation. The proper lifetime is the time interval that would be measured in the rest frame of the particle, which is different from the time measured in the detector's frame due to the particle's high speed.

Understanding Time Dilation

According to Einstein's theory of relativity, time measured in a moving frame (the detector's frame in this case) is different from the time measured in the rest frame of a particle. This effect is quantified by the Lorentz factor, denoted as γ (gamma).

The Lorentz Factor

The Lorentz factor is calculated using the following formula:

• γ = 1 / √(1 - (v²/c²))

Here, v is the speed of the particle, and c is the speed of light. In our scenario, the speed of the particle is 0.992c. Let's calculate γ:

First, we square the speed:

• v² = (0.992c)² = 0.984064c²

Now, we can find (1 - v²/c²):

• (1 - v²/c²) = 1 - 0.984064 = 0.015936

Next, we take the square root:

• √(0.015936) ≈ 0.1263

Now we can find γ:

• γ = 1 / 0.1263 ≈ 7.91

Calculating the Time Dilation

In the detector's frame, the lifetime of the particle (the dilated lifetime) can be expressed as:

• Δt = γ * Δt₀

Where Δt₀ is the proper lifetime that we want to find. We can rearrange this to solve for Δt₀:

• Δt₀ = Δt / γ

Finding the Dilated Lifetime

To find the dilated lifetime (Δt), we can use the distance traveled by the particle before it decays:

• Distance = Speed * Time

We have a distance of 1.05 mm and speed of 0.992c. First, we convert 1.05 mm to meters:

• 1.05 mm = 0.00105 m

Using the speed of light in meters per second (approximately 3.00 x 10^8 m/s), we can calculate the dilated lifetime:

• Speed = 0.992 * 3.00 x 10^8 m/s = 2.976 x 10^8 m/s

Now, we can find the time:

• Δt = Distance / Speed = 0.00105 m / (2.976 x 10^8 m/s) ≈ 3.52 x 10^-9 s

Calculating the Proper Lifetime

Now, we can substitute Δt into the equation for proper lifetime:

• Δt₀ = Δt / γ

Plugging in the values:

• Δt₀ = (3.52 x 10^-9 s) / 7.91 ≈ 4.45 x 10^-10 s

Thus, the proper lifetime of the unstable high-energy particle, had it been at rest with respect to the detector, is approximately 4.45 x 10^-10 seconds. This demonstrates how high speeds can significantly alter the perception of time due to relativistic effects.