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How can the time dilation equation explain faster moving clocks running slower?

Problem Statement
Samir (who is standing on the ground) starts his stopwatch at the instant that Maria flies past him in her spaceship at a speed of 0.600c. At the same instant, Maria starts her stopwatch. As measured in Samir's frame of reference, what is the reading on Maria's stopwatch at the instant that Samir's stopwatch reads 10.0s?
Relevant Equations
delta t = delta t0/sqrt(1-(v/c)^2)
Since for the two events of Samir starting the stopwatch, and the stopwatch reaching 10.0s, Samir and his stopwatch are stationary from his own frame of reference, I said it was the proper time and that delta t0 = 10s. Then the speed of the moving frame of reference was 0.6c. I thought placing this in the given time dilation equation should give the time on Maria's stopwatch, as I thought it would be the time passed from Maria's frame of reference, but it gives delta t = 12.5 seconds. This does not match the idea that a clock observed to be moving at relativistic speeds should tick slower.
I feel as though I am missing something simple, but I can't comprehend why the clock would tick slower when delta t is always larger than or equal to delta t0 according to this equation. Any help would be much appreciated.
 

PeroK

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Time dilation in this case is symmetric. In Samir's frame, Maria's clock is running slow. But, in Maria's frame it is Samir's clock that is running slow.
 
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If Samir's clock advances only 10 seconds during the time interval, as reckoned from the ground frame of reference, that their clocks have advanced 12.5 seconds, doesn't that mean that they would conclude that Samir's clock is running slow?

Maria can no longer see Samir by the time he has traveled 6 light-seconds. So she has to depend on Jose, who is stationed on the ground a distance of 6 light-seconds away. Maria and Jose have clocks that they have synchronized with one another (in advance). Each writes down on a piece of paper the time on their clocks at which Samir went past them. Then, later, they get together and compare notes. They find that, according to their synchronized clocks, 12.5 seconds have elapsed between the time he passed Maria and the time he passed Jose. They also noted the time that his clock indicated as he went past each of them. When the compare notes, they determine that his clock only showed a change of 10 seconds between the time he passed Maria and the time he passed Jose.
 
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Thank you for your replies, but I still don't quite understand. If from Samir's frame of reference, his clock advances 10 seconds and Maria's advances 12.5 seconds, isn't Maria's clock running fast, when it should be slow from Samir's view? According to my textbook, Samir should have observed that Maria's clock advanced 8 seconds, although it does not explain why. Is delta t not the value Maria's clock displays from Samir's point of view in this case?
 

PeroK

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Thank you for your replies, but I still don't quite understand. If from Samir's frame of reference, his clock advances 10 seconds and Maria's advances 12.5 seconds, isn't Maria's clock running fast, when it should be slow from Samir's view? According to my textbook, Samir should have observed that Maria's clock advanced 8 seconds, although it does not explain why. Is delta t not the value Maria's clock displays from Samir's point of view in this case?
You may simply be multiplying instead of dividing by the gamma factor; or, vice versa.

In Samir's frame, Maria's speed is ##0.6c##, hence the gamma factor is ##\gamma = \frac{1}{\sqrt{1- v^2/c^2}} = 1.25##. To get the time on Maria's clock (as measured in Samir's fame), you divide by ##\gamma##:

##\tau = \frac{t}{\gamma} = t \sqrt{1- v^2/c^2}##
 
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Thank you for your replies, but I still don't quite understand. If from Samir's frame of reference, his clock advances 10 seconds and Maria's advances 12.5 seconds, isn't Maria's clock running fast, when it should be slow from Samir's view? According to my textbook, Samir should have observed that Maria's clock advanced 8 seconds, although it does not explain why. Is delta t not the value Maria's clock displays from Samir's point of view in this case?
If there is another guy in Samir's frame of reference that is located 6 light seconds behind Samir (and his clock is synchronized with Samir), he will determine that, when Maria goes past him, only 8 seconds have elapsed since she had passed Samir.
 
Okay, I understand a bit more now - thank you both for your time :)
 

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