Question

The negative pion (π−) is an unstable particle with an average lifetime of 2.60×10−8s (measured in the rest frame of the pion).

If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be 4.20×10−7s. Calculate the speed of the pion expressed as a fraction of c
What distance, measured in the laboratory, does the pion travel during its average lifetime?

Answer 1

The average speed of the pion is approximately 0.997 times the speed of light (c). The pion travels a distance of approximately 126 meters during its average lifetime, as measured in the laboratory frame.

Step-by-step explanation:

The average lifetime of a negative pion (π−) is 2.60×10−8 seconds in its rest frame and 4.20×10−7 seconds in the laboratory frame when the pion is traveling at a very high speed relative to the laboratory. To calculate the speed of the pion, we can use the time dilation formula, which states that the ratio of the average lifetimes in the two frames is equal to the Lorentz factor γ. Using this formula, we find that γ = (4.20×10−7)/(2.60×10−8) ≈ 16.15. The speed of the pion can then be calculated as a fraction of the speed of light (c) using the formula γ = 1/√(1 - (v/c)²). Solving for v, we find that v/c ≈ √(1 - 1/γ²) ≈ 0.997, or approximately 0.997c.

To calculate the distance traveled by the pion during its average lifetime, we can use the formula d = ct, where d is the distance, c is the speed of light, and t is the average lifetime measured in the laboratory frame. Substituting the given values, we have d = (3.0x10⁸ m/s)(4.20×10−7 s) ≈ 126 meters.

Answer 2

The speed of the pion relative to the speed of light is approximately 0.872. The distance traveled by the pion in the laboratory during its average lifetime is approximately 3.3x10⁻¹⁶ meters.

Step-by-step explanation:

To calculate the speed of the pion, we need to use the time dilation formula. The time dilation formula relates the observed time of an event to the time in the rest frame of the object:

t'=t/sqrt(1-v²/c²)

where t' is the observed time, t is the rest frame time, v is the speed of the pion, and c is the speed of light.

Using the given values, we can plug in the numbers and solve for v:

4.20x10⁻⁷s=2.60x10⁻⁸s/sqrt(1-v²/c²)

Simplifying the equation and solving for v, we find that v/c is approximately 0.872.

To calculate the distance travelled by the pion during its average lifetime in the laboratory, we can use the equation d=vt, where d is the distance, v is the speed, and t is the time. Substituting the values, we find that the distance is approximately 3.3x10⁻¹⁶ meters.