Question

After being created in a high-energy particle accelerator, a pi meson at rest has an average lifetime of 2.60 10-8 s. Traveling at a speed very close to the speed of light, a pi meson travels a distance of 110 m before decaying. How fast is it moving? (Enter your answer to four significant figures.)

Answer 1

First we need to calculate the time dilation,

t' = t/sqrt(1 - (v/c)²)

c=speed of light

velocity,v = s/t

The problem here is to find out, by what factor the pions proper time is dilated in order for it to be able to travel 110 m. The time required for travelling 110 m is:

t' = s/v

which is a lot larger than the pions life-time in its rest-frame. With the help of the first formula we can express t' in terms of the pions life-time t:

t/sqrt(1 - (v/c)²) = s/v

We have all the information needed to solve this equation. First, we square it to get

v² * t²/s² = 1 - v²/c²

(t²/s² + 1/c²) * v² = 1

v² = 1/(t²/s² + 1/c²)

which can be simplified to, |v| = 1/sqrt(t²/s² + 1/c²)

or

|v| = 1/sqrt(1 + c² * t²/s²) * c

This equation can used to find the percentage of the velocity in relation to the speed of light.

Now you can either use this exact formula or expand the expressions a la Taylor with the help of these partial expansions:

sqrt(1 + x) ~1 + x/2 + O(x²)

1/(1 + y) ~ 1 - y + O(y²)

which is possible, because the value of c²*t²/s² is rather small. By simplifying again we get,

|v| ~ (1 - 0.5 * c² * t²/s²) * c

v=(1-(0.5*(3*10⁸)²*(2.6*10^-8)²/110²)3*10⁸=0.99 c=2.992*10⁸ m/s