Hello, it's me
I was wondering if after all these years you'd like to meet
I was wondering if you guys could explain how to determine which time is the dilated time and which time is the proper time (Einstein's special theory of relativity) when given questions TT^TT So for like this one:
"A spaceship travelling through space experiences a 90% time dilation. How fast is it travelling?"
I thought the dilated time was 0.9 and proper time was 2, but then you'd eventually have to square root a negative so that doesn't work out :/
And for this one
"The high-speed muons produced for an experiment by the Fermilab accelerator are measured to have a lifetime of 5.0 microseconds. When these muons are brought to rest, their lifetime is measured to be 2.2 microseconds"
I thought you could look from the perspective of the muon (since relative motion, doesn't matter whether you look from muon or lab) so you'd have proper time to 5 microseconds and dilated time to be 2.2 but apparently not :/ that too gives you a square root of a negative number. But the other way doesn't make sense to me, if proper time is 2.2 seconds and dilated time is 5 seconds, won't that mean in 2.2 seconds, 5 seconds passed for the muon? Meaning each second is faster idk
So basically I was wondering how you tell which one is proper and which one is dilated 
Love y'all,
Neutron
Hey Neutron! This confused the heck out of me in Year 12 as well! I'll start by explaining the two examples you provided, then give you the trick to make sure you have everything in the right direction
So, your first example. I would interpret "90% time dilation" as time passes by at 10% of the normal rate. That is, the dilated time is 10 times greater than the undiluted time. A spaceship travelling extremely quickly will have time pass by much slower, as measured from earth. I could be misinterpreting slightly, but in any case, we have some kind of effect on time passage.
Remembering our time dilation formula, I'll refer to the time term on the LHS as dilated time, and the RHS as unaffected time.
In this case, the
dilated time is 10 times greater than the actual, unaffected time. Therefore:
This one is a little hard to understand, the next one is a bit easier.
In your second example, 5.0 microseconds is the dilated time. This is what we measure when the particle is at speed. It moving affects our time measurements. Then, the 2.2 microseconds is the unaffected time. Your statement about "if proper time is 2.2 seconds and dilated time is 5 seconds, won't that mean in 2.2 seconds, 5 seconds passed for the muon", is correct! What the muon would experience in 5 seconds only takes us 2.2 seconds to experience. The issue with your approach is that the time measurements are made from the earths frame of reference, not the muons. So you have to go from the earth as unaffected time. That is what had you a bit backwards.
If you are at all confused, your check is simple.
Dilated time must ALWAYS be larger than unaffected time. Time dilation causes time to pass by slower, so, the dilated time difference is larger. This is an easy way to check your reasoning against the question.
I hope this helps! This whole thing is a little confusing, your dilated time represents the time as measured by a stationary observer. It will thus always be larger
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