Calculating Muon Velocity with Lorentz Transformation | Physics Homework

[kd001]

Homework Statement

Muons, which have a half-life of 2 x 10-6 s, are formed in the Earth's atmosphere at an
altitude of 10 km. If they travel normal to the Earth's surface, and one half of them
reach it before they decay, what is their velocity?

Homework Equations

Lorentz Transormation.

The Attempt at a Solution

My understanding of the question is that in the reference frame of the the muons 2 x 10-6 s elapses before they reach the surface of the Earth. So t' is 2 x 10-6 s. So I need to find out the time that elapses in the Earth's reference frame in order to calculate the relative velocity. However, the only other information I've got is x=10km (taking x to be the direction of the velocity of the muons). Aren't there too many unknowns?

 

[tiny-tim]

Hi kd001!

My understanding of the question is that in the reference frame of the the muons 2 x 10-6 s elapses before they reach the surface of the Earth. So t' is 2 x 10-6 s. So I need to find out the time that elapses in the Earth's reference frame in order to calculate the relative velocity. However, the only other information I've got is x=10km (taking x to be the direction of the velocity of the muons). Aren't there too many unknowns?

Call the velocity v.

Find the time t (in the Earth frame), and then see if t/t' agrees with that velocity.

 

[kd001]

I don't see how t can be calculated without knowing v in the first place.

 

[tiny-tim]

I don't see how t can be calculated without knowing v in the first place.

Find t as a function of v …

what do you get? ​

 

[paranormal]

I would first clarify any uncertainties about the given information and the question. For example, I would confirm whether the given half-life of muons is for their rest frame or for the Earth's frame. I would also clarify if the muons are traveling at a constant speed or if they are accelerating. Additionally, I would ask for any other relevant information, such as the distance traveled by the muons, to ensure an accurate calculation.

Assuming the given information is correct, we can use the Lorentz transformation to calculate the velocity of the muons. However, as you have pointed out, there are not enough known variables to directly calculate the velocity. We need to first find the time elapsed in the Earth's frame, which is the time dilation factor, in order to calculate the relative velocity.

To find the time dilation factor, we can use the equation t' = γt, where t' is the time elapsed in the Earth's frame, t is the time elapsed in the muon's frame (given as 2 x 10-6 s), and γ is the Lorentz factor. We can rearrange this equation to find γ = t'/t.

Next, we can use the Lorentz transformation equation for velocity, v = (v'+u)/(1+uv'/c^2), where v is the velocity of the muons in the Earth's frame, v' is the velocity of the muons in their own frame (which we want to find), u is the velocity of the Earth's frame (which we assume to be zero since the muons are traveling normal to the Earth's surface), and c is the speed of light.

Substituting the known values, we get v = (v' + 0)/(1+0v'/c^2). Simplifying, we get v = v', which means the velocity of the muons in their own frame is equal to the velocity of the muons in the Earth's frame. This makes sense since the muons are traveling at a constant speed and there is no acceleration.

Therefore, the velocity of the muons is 10 km/2 x 10-6 s = 5 x 10^9 m/s or approximately 0.99c (c = speed of light). This is a very high velocity, which is expected for particles with a short half-life.

In conclusion, by using the Lorentz transformation