Length, time and velocity in relativity (special?)

[weirdobomb]

10. The average lifetime of a pi meson in its own frame of reference is 2.6 x 10^-8 s. If the meson
moves with a speed of 0.95c, what is (a) its mean lifetime as measured by an observer on Earth and
(b) the average distance it travels before decaying, as measured by an observer
on Earth?

(a) using the formula for time dilation
mean lifetime measured on Earth I found t = 8.12 x 10^-4 s

(b) I calculated the distance traveled in the particle to travel in its own frame L = 7.44 m
I took that value and used the formula for length contraction
average distance traveled measured on Earth before decaying L = 23.8 m.

I am pretty sure my answers are correct, or at least the concept is. My question is for (b), how do I find the velocity of the particle measured on Earth? What I want to do is take the Earth measured velocity to multiple by the answer from (a) to have a different way to answer (b).

I think I have to use the velocity formula in the attachment but doesn't that one require the particle to be moving in an already moving frame (v=velocity of frame u'=particle velocity in that frame).

 

[Doc Al]

(a) using the formula for time dilation
mean lifetime measured on Earth I found t = 8.12 x 10^-4 s

Show what you did here. What value did you get for the Lorentz factor (gamma)?

 

[weirdobomb]

I correct my answer to t = 8.33 x 10^-8 s which screws up everything. y = 3.2

I don't care about the answers for (a) and (b). I really want to know how to find the velocity for particle measured on Earth.

 

[Doc Al]

I don't care about the answers for (a) and (b). I really want to know how to find the velocity for particle measured on Earth.

That's given:

If the meson moves with a speed of 0.95c,

That's its speed with respect to the Earth. (Its speed with respect to itself is zero, of course.)

 

[HalfLostAndSearching]

Can I just use the velocity formula without the frame velocity?

As a scientist, it is important to have a clear understanding of the concepts and formulas being used in a problem. In this case, we are dealing with the concepts of length, time, and velocity in the context of special relativity. Special relativity is a theory that describes how physical laws and measurements change when objects are moving at speeds close to the speed of light.

In special relativity, the concept of time is relative and depends on the observer's frame of reference. This means that different observers will measure different times for the same event, depending on their relative motion. The same applies to the concept of length, which also appears to change depending on the observer's frame of reference.

In this particular problem, we are dealing with a pi meson moving at a speed of 0.95c (95% of the speed of light) and its average lifetime in its own frame of reference, which is 2.6 x 10^-8 s. We are asked to find the meson's mean lifetime and the average distance it travels before decaying, as measured by an observer on Earth.

(a) To find the mean lifetime as measured by an observer on Earth, we can use the formula for time dilation, which states that the time measured by an observer in a moving frame is longer than the time measured in the stationary frame. This means that the meson's lifetime will appear longer when measured by an observer on Earth. Using the given values, we can calculate the mean lifetime as t = 8.12 x 10^-4 s.

(b) To find the average distance traveled before decaying, we need to use the concept of length contraction, which states that the length of an object appears shorter when measured in a moving frame than in a stationary frame. Using the formula for length contraction, we can calculate the average distance traveled by the meson as L = 23.8 m.

Now, to answer your question about finding the velocity of the particle measured on Earth, we can use the velocity formula without the frame velocity. This formula, v = u/(1-v^2/c^2)^0.5, is used to find the velocity of an object in a stationary frame when its velocity is known in a moving frame. In this case, we know the velocity of the meson in its own frame (u) and we want to find its velocity in the stationary frame (