How Do Firecracker Explosions and Pion Decay Demonstrate Special Relativity?

[physicsug]

Homework Statement

Q1. Firecrackers are placed at points A and B, which are 100m apart as measured in the rest frame of the earth. As a rocket shop, moving at speed v = 3/5 c, passes point A the firecracker there explodes. As the rocket passes point B, the second firecracker explodes. As read on the rockets clock, the time difference between the explosions was delta te. Clocks synchronized in the rest frame of the Earth measure a time interval delta te between the explosions.

a) Determine delta te.
b) Determince delta te by reasoning using time dilation , and again by reasoning using length contraction. Do your results agree? Should they?


Q2. Neutral pions decay into two gamma rays. In its frame of reference, a neutral pion has a lifetime of 8.4 x 10^-17s (it is effectively a clock which ticks once). A particle accelerator produces pions with a speed of 0.99975c. What is the lifetime of the pions measured in the laboratory frame? How far do the pions move in that time?

2. Homework Equations
Lorentz transformation for time and length.

The Attempt at a Solution

For Q1:
a) I do not understand why both measured times are delta te. Is it because the times are measured in their respective frames?
Taking delta te to be the proper time in the rest frame of the earth,
I calculated delta te = 100/(3/5 c)

b) I do not understand what the question is asking. I'm not sure how to approach the question but I think that the results should agree as time dilation should be consistent with the length contraction depending on the frame of reference, am I correct?

For Q2:
It says in the question "In its frame of reference, a neutral pion has a lifetime of 8.4 x 10^-17s", does this mean that the lifetime given is the proper time since it is in the frame of reference of the pion?


From what I understand, for the same event, the time measured in the respective time frame is the same. Its only when one frame is viewed from another that there is a difference in measured time and length. Are these correct?

I am still new to this, all help appreciated. Thank you.

 

[Doc Al]

For Q1:
a) I do not understand why both measured times are delta te. Is it because the times are measured in their respective frames?

I suspect there is a typo in your problem statement--double check it. I'd call one time interval t_earth and the other t_rocket.

For Q2:
It says in the question "In its frame of reference, a neutral pion has a lifetime of 8.4 x 10^-17s", does this mean that the lifetime given is the proper time since it is in the frame of reference of the pion?

Yes.

 

[physicsug]

I suspect there is a typo in your problem statement--double check it. I'd call one time interval t_earth and the other t_rocket.

I thought that it was a typo as well, but apparently not.
Assuming one of them is t_rocket:
using time dilation, t_rocket = 6.944 x 10^-7 s
using length contraction, t_rocket = 4.444 x 10^-7 s
Is that correct?

 

[Doc Al]

I thought that it was a typo as well, but apparently not.

As long as you realize that there are two different time intervals involved.

Assuming one of them is t_rocket:
using time dilation, t_rocket = 6.944 x 10^-7 s

Describe how you found this. (I think you used the time dilation equation backwards.)

using length contraction, t_rocket = 4.444 x 10^-7 s

That looks good.

Is that correct?

Both methods should give the same answer.

 

[physicsug]

I get it now. Thank you for your help.

 

[physicsug]

Going by the definition, proper time is measured by the clock which position remains fixed in a frame. And the dilated time is always greater than the proper time. So based on the answer, t_rocket is then the proper time and t_earth is the dilated time. Is that right?

 

[Doc Al]

Going by the definition, proper time is measured by the clock which position remains fixed in a frame. And the dilated time is always greater than the proper time. So based on the answer, t_rocket is then the proper time and t_earth is the dilated time. Is that right?

Here's how I would put it. Clocks always measure proper time. As far as the rocket frame goes, the two events are co-located with the rocket clock, thus t_rocket represents a proper time. But in the Earth frame the two events are not co-located, and two synchronized clocks are used to measure t_earth, so t_earth is not a proper time measured by a single clock. (According to the rocket frame, those two Earth clocks are not synchronized.)

 

[physicsug]

Alright, thank you for explaining.

 

[physicsug]

I have another question:

Homework Statement

The quantity I = -(ct)² + x² + y² + z² is called the "invariant interval" because it is unchanged under Lorentz transformations. In other words the distance between an event and the origin changes and the time changes, but I doesn't. Prove this is true by applying a Lorentz transformation to I.

Homework Equations

Lorentz transformation

The Attempt at a Solution

Not really sure how to apply Lorentz transformation on I. I tried with some substitions on x, y and z but it made little sense.

 

[Doc Al]

Replace t, x, y, & z by their Lorentz transforms so that you express the interval in terms of primed quantities only. See what happens.

 

[physicsug]

Thanks. I couldn't get it before because I use the Lorentz transformation on x for y and z as well.

The Lorentz transformation for y and z are y = y' and z = z' respectively. Are they so because the direction of motion considered is taken to be the x direction since space is isotropic? And the fact that velocity addition in the x,y and z direction are different, is it merely due to the Lorentz transformation of x,y and z as the velocities are the derivatives of the transformation?