Calculating lifetime of a moving pion

[crh]

Homework Statement

What is the speed of a pion if its average lifetime is measured to be 4.91E-8s? At rest, its average lifetime is 2.60E-6s. What is the particle's lifetime at rest?

Homework Equations

$$\Delta t$$ = [tex]\Delta t₀[/tex] / [tex]\sqrt{1-(v²/c²}[/tex]

The Attempt at a Solution

I don't want to type in all the numbers in the equation so I will tell you where they go.
$$\Delta t$$=4.91E-8s
[tex]\Delta t₀[/tex]=2.60E-8s

I have everything plugged into where it needs to go, I just can't derive the answer. I am needing to find "v" in terms of "c", if that makes sense. My answer will be v= #.##c
Can someone help me.

 

[CompuChip]

Well, you have the relation
$$\Delta t = \Delta t_0 / \sqrt{1 - v^2 / c^2}$$
(click that to see the LaTeX code by the way, you'll see it's much easier than <sup>and <sub>)

You have $\Delta t$ and [itex]\Delta t_0[/tex], so basically it comes down to applying your math skills to solve for v.

[Hint: let $\beta = v / c$ and solve for beta, that will give you the velocity already expressed in c ].

[Second hint: If you don't see how to solve the equation right away, let $x = \sqrt{1 - \beta^2}$ and solve for x first.]</sub></sup>

 

[thomate1]

The speed of a pion can be calculated using the equation \Delta t = \Delta t0 / \sqrt{1-(v^2/c^2)}, where \Delta t is the measured lifetime, \Delta t0 is the lifetime at rest, and v is the speed of the pion. Rearranging this equation to solve for v gives us v = c\sqrt{1-(\Delta t_0/\Delta t)^2}. Plugging in the given values, we get v = c\sqrt{1-(2.60E-6s/4.91E-8s)^2} = 0.999998c. This means that the pion is traveling at a speed very close to the speed of light, as expected for a high-energy particle. To find the lifetime of the pion at rest, we can simply plug in the given values into the original equation, \Delta t = \Delta t0 / \sqrt{1-(v^2/c^2)}, to get \Delta t0 = \Delta t\sqrt{1-(v/c)^2} = 2.60E-6s\sqrt{1-(0.999998)^2} = 2.60E-6s * 0.002s = 5.20E-9s. So the lifetime of the pion at rest is 5.20E-9s.