Question

If a charged pion that decays in 10⁻⁸ second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion’s speed must be most nearly

a) 3 × 10⁸ m/s
b) 1.5 × 10⁸ m/s
c) 6 × 10⁸ m/s
d) 0.75 × 10⁸ m/s

Answer 1

In the laboratory, the charged pion must travel a distance of 30 meters before decaying. The time during which it exists can be calculated using the equation: d = ct. Rearranging the equation, we have t = d/c. Plugging in the values, we get t = (30 m) / (3.0 x 10^8 m/s). Solving this equation gives us the time of approximately 1.0 x 10^-7 seconds.

This correct answer is a)

Step-by-step explanation:

In the laboratory, the charged pion must travel a distance of 30 meters before decaying. The time during which it exists can be calculated using the equation: d = ct, where d is the distance, c is the speed of light, and t is the time.

Rearranging the equation, we have t = d/c. Plugging in the values, we get t = (30 m) / (3.0 x 10^8 m/s). Solving this equation gives us the time of approximately 1.0 x 10^-7 seconds.

Now, in the pion's own rest frame, it decays in 10^-8 seconds. This means that the pion's speed must be derived from the time dilation equation: t' = t / √(1-v^2/c^2). Rearranging this equation, we have v = c - (d' / t'), where v is the speed, c is the speed of light, d' is the distance traveled, and t' is the time in the rest frame.

Plugging in the values, we get v = (3.0 x 10^8 m/s) - (30 m / 10^-8 s). Solving this equation gives us the speed of approximately 2.7 x 10^9 m/s. Therefore, the pion's speed must be most nearly option a) 3 x 10^8 m/s.

This correct answer is a)