Speed of Pi Meson - Relativistic question

[ZedCar]

After being created in a high-energy particle particle accelerator, a pi meson at rest has an average lifetime of 2.60 x 10^-8s. Travelling at a speed very close to the speed of light, a pi meson travels a distance of 120m before decaying. How fast is it moving?

Answer: 0.998c

Could anyone give me an ideas as to how to start this, or which equation(s) I should be using?

Thank you

 

[niklaus]

First calculate the pion life time in the lab frame. It's ok to use v = c here.

Then all you need is time dilatation and how the gamma factor depends on v.

 

[Doc Al]

Could anyone give me an ideas as to how to start this, or which equation(s) I should be using?

Hint: You are given ΔT₀. What would the lifetime be in the lab frame? Write it in terms of the unknown speed v. (Do not assume v = c!) You'll then set up an equation and solve for the speed.

 

[ZedCar]

Hint: You are given ΔT₀. What would the lifetime be in the lab frame? Write it in terms of the unknown speed v. (Do not assume v = c!) You'll then set up an equation and solve for the speed.

I would use the time dilation formula to obtain the lifetime in the lab frame.

Δt = γΔt0

Δt = γ(2.60x10^-8)
So I would have two unknowns, Δt and the v value in γ.

Is this correct so far? I solve this for v, which means the expression will have an unknown of Δt contained in it.

 

[cupid.callin]

yes you're correct ... i just solved it this way and it gives the correct answer ...

 

[Doc Al]

Is this correct so far? I solve this for v, which means the expression will have an unknown of Δt contained in it.

So far, so good. Now combine this with the basic equation for velocity, which will eliminate Δt. Then you can solve for v, which will be the only unknown.

FYI, the method suggested by niklaus, of calculating Δt by taking v = c, is perfectly fine for this problem. The error introduced by that short cut is too small to matter.

But you should have no problem solving for v without needing any approximations.