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

In summary: Yes, delta t is not the value that Maria's clock displays from Samir's point of view in this case. The correct value would be delta t0.
  • #1
nopenopedynope
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Homework 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.
 
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  • #2
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?
 
  • #5
nopenopedynope said:
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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  • #6
nopenopedynope said:
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.
 
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  • #7
Okay, I understand a bit more now - thank you both for your time :)
 

1. What is the time dilation equation?

The time dilation equation, also known as the time dilation factor, is a mathematical formula that describes how time passes differently for objects in different frames of reference, specifically when one object is moving at a high velocity relative to the other. It is represented by the symbol γ (gamma) and is calculated by dividing the time experienced by the moving object by the time experienced by the stationary object.

2. How does the time dilation equation explain faster moving clocks running slower?

The time dilation equation helps to explain this phenomenon by showing that as an object's velocity increases, its time dilation factor also increases. This means that time will appear to pass slower for the faster-moving object, relative to a stationary observer. This is because the faster an object moves, the more distance it covers in a given amount of time, which leads to a dilation or stretching of time.

3. Can the time dilation equation be applied to any moving object?

Yes, the time dilation equation can be applied to any moving object, as long as its velocity is high enough to have a noticeable effect on time. However, the effects of time dilation are most noticeable at speeds close to the speed of light, which is why it is often studied in the context of objects traveling at extreme velocities, such as in space travel or particle accelerators.

4. What are some real-world applications of the time dilation equation?

One of the most well-known applications of the time dilation equation is in the theory of relativity, which is a fundamental concept in physics. It also has practical applications in fields such as space travel and satellite technology, where precise measurements of time are crucial. Additionally, the time dilation equation is used in particle accelerators to account for the effects of time dilation on the measurements of subatomic particles traveling at high speeds.

5. Are there any limitations to the time dilation equation?

While the time dilation equation is a useful tool for understanding the effects of time dilation, it does have some limitations. It only applies to objects moving at constant velocities in uniform motion. It also does not take into account other factors that can affect time, such as gravitational forces. Additionally, the time dilation equation is a theoretical concept and has not been proven to be 100% accurate in all situations.

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