Calculating Average Lifetime of Particles at Rest

[msimard8]

Heres the question before I say anything else

A beam of unknown elementary particles travels at a speed of 2.0 x 10^6 m/s. Their average lifetime in th ebeam is measrured to be 1.6 x 10^-8 s. Caclulate their average lifetime when at rest.

hmm I have no examples to go by. I figure I will have to use the formula

tm = ts/[sqrt(1-v^2/c^2)]

where tm is the time it takes when the object is moving (for example when viewing from Earth and ts is the stationary time, v is speed of the object and c is speed of light)

I tried to do this by plugging in the numbers as

ts=?
v=2.0x10^8 m/s
tm=1.6x10^-8

I didn't get the right answer of 1.2 x 10^-8 s so I am assuming I am wrong.
How do I do this correctly.

 

[Kurdt]

Well you are using the right formula and I get the right answer when I plug the numbers in. If you tell us what answer you get perhaps we can tell you what you're doing wrong.

EDIT: just noticed you cited

I tried to do this by plugging in the numbers as

ts=?
v=2.0x10^8 m/s

as what you've input.

That could be the problem.

 

[kyle8921]

I would first clarify the units being used in the question. The speed given is 2.0 x 10^6 m/s, while the average lifetime is measured in seconds. This discrepancy may affect the calculation.

To calculate the average lifetime at rest, we can use the formula for time dilation:

t_0 = t_m/sqrt(1-(v^2/c^2))

Where t_0 is the time at rest, t_m is the time when the particles are moving, v is the speed of the particles, and c is the speed of light.

Plugging in the values given in the question, we get:

t_0 = (1.6 x 10^-8 s)/sqrt(1-(2.0 x 10^6 m/s)^2/(3.0 x 10^8 m/s)^2)

Simplifying this, we get t_0 = 1.6 x 10^-8 s/sqrt(1-0.000004) = 1.2 x 10^-8 s.

Therefore, the average lifetime of the particles at rest is 1.2 x 10^-8 seconds. It is important to note that this is an average value and individual particles may have slightly different lifetimes. Also, it is worth mentioning that this calculation assumes that the particles are moving at a constant speed and do not experience any acceleration or deceleration.