How to Calculate Speed and Time for Pions and Spacecraft?

[jlmac2001]

1. What is the speed of a beam of pions if their average lifetime is measured to be 4.34×10-8s? At rest, their lifetime is 2.60×10-8s.

- Would I use velocity= distance/time? 4.34/2.6=velocity

2. A certain star is 70.6 light-years away. How many years would it take a spacecraft traveling 0.910c to reach that star from Earth, as measured by observers on Earth? How many years would it take to reach that star from Earth, as measured by observers on the spacecraft ?

- For the first part, would it be 70.6/sqrt(1-.910)?
-For the second part, would it be the answer from part 1 divided by sqrt(1-.910)?

 

[HallsofIvy]

I hate to say it but you need serious help:
"Would I use velocity= distance/time? 4.34/2.6=velocity"
4.34 is given as the "lifetime"- it is time not distance!
You seem to be just grabbing the two numbers and putting them together is the simplest way. Considering that in order to do this problem you have to be studying relativity, you will need to be able to actually understand things better than that!
If something has a longer lifetime (4.34 is longer than 2.6) at higher speed, it has to be "living" slower. Do you know anything about speed causing time to slow down?

Looking at your remarks about problem two, I'm a little more reassured. Since the Earth is not moving, the time required from the Earth viewpoint for the rocket (at .91c) to go a distance 70.6 light year is just the usual formula: speed= distance/time so time= distance/speed: 70.6/0.91 years (since a "light year" is defined as the distance light goes in a year, we don't have to worry about "c").

From the point of view of the people in the rocket (and since both distance and speed given are measured relative to the earth), we have to allow for the "slowing" of time: but the formula would be 1/(1- 0.91^2).

 

[M98Ranger]

1. Yes, you would use the formula velocity = distance/time. However, in this case, the distance is not given. Instead, you need to use the fact that the average lifetime of the pions is equal to the distance traveled at the speed of light. So, the velocity would be 2.6/4.34 = 0.599c.

2. For the first part, you are correct. The formula for time dilation is t' = t/sqrt(1-v^2/c^2), where t is the time measured by the observer on Earth and t' is the time measured by the spacecraft. So, in this case, t' = 70.6/sqrt(1-0.910^2) = 70.6/0.420 = 168.1 years.

For the second part, you would use the same formula, but with the velocity of the spacecraft (0.910c) instead of the Earth observer (0.599c). So, t' = 70.6/sqrt(1-0.910^2) = 70.6/0.420 = 168.1 years. This means that both observers would measure the same amount of time for the spacecraft to reach the star.