Hello! I have to hand in this coursework tomorrow and I just wanted to check I had done it correctly so if someone could have a look I'd be really grateful... It's quite long...

## Homework Statement

A negatively charged pion $$\pi$$$$^{-}$$ has mass m=140MeV/c$$^{2}$$ and lifetime 2.6 x 10$$^{-8}$$s.

a) If the pion is accelerated with respect to the laboratory such that it has a total energy of 2.1 x 10$$^{3}$$ MeV, show that the relativistic factor is 15.
b) Using this factor, determine the pion lifetime when measured in the laboratory frame.
c) Whar is the speed of the pions (in units of c) in the laboratory frame?
d) At time t=0, 2000 of these energetic pions are produced in the laboratory. How many pions will be left after 1 $$\mu$$s, as measured on a clock in the laboratory?
e) What is the mean distance in the lab travelled by the pions before they decay?

## Homework Equations

E = $$\gamma$$mc^2
T$$_{0}$$ = t/$$\gamma$$

## The Attempt at a Solution

Ok here is what I have done...

a) E = $$\gamma$$mc^2

2.1 x 10$$^{9}$$ = $$\gamma$$ (140 x 10^6)

therefore gamma is 15...

b) T0 = t/$$\gamma$$

= (2.6 x 10^8)/15

T0 = 1.0 x 10^-9s

c) 1.0 x 10^-9 = (2.6 x 10^8)/$$\gamma$$

1/(1 - (v^2/c^2))^1/2 = (2.6 x 10^8)/$$\gamma$$

rearranging and simplifying...

I got v = 0.998c

d) This is where I got stuck, I'm not sure which formula to use or how to go about this.
e) Same for this question.

Any help will be greatly appreciated :)