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Anderson Video - Time Dilation

Professor Anderson
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Well, one thing we're worried about is time. And we talked about this in class. There is something called time dilation. In words, it's very simple. Moving clocks run slow. In mathematical terms, it's written as the following. Delta t equals delta t naught over the square root 1 minus V squared over C squared. But what are these different things? Right? If you're here on the Earth and you're sitting here with your clock you're gonna measure delta t. Somebody flying by in the spaceship they're gonna measure delta t naught. And those two time intervals don't sync up. Okay, they're not the same, they are scaled by this factor gamma, which is 1 over the square root of 1 minus V squared over C squared. Okay, so we talked about this in class. Let's take a look at one of your homework problems that deals with this. Okay. And this is problem 26.02. So it says the following. We have an elementary particle. This might be like a muon. Muon are particles that are streaming through our atmosphere. They come from the Sun. And they have a very short lifetime. The lifetime of this thing delta t is 3.86 microseconds, which is 10 to the minus 6 seconds. But the speed is pretty fast. For mine I have 2.55 times 10 to the 8 meters per second. So it's a good fraction of the speed of light. And in this question we need to answer, "what is delta t naught?" In other words, the particle is now the person in the spaceship flying past. The lifetime is how long is that particle alive as measured on their clock. What we measure is delta t. Okay, what we measure here on Earth. All right. It almost looks like we have everything we need, right? We just need to take this equation and solve for dela t naught. So, delta t naught is going to be if I just multiply across by this square root that should do it. 1 minus V squared over C squared. And now we can multiply it out. We've got delta t, which is right here. 3.86 times 10 to the minus 6 seconds. I've got the square root of 1 minus 2.55 times ten to the 8. I'm going to divide by 3 times 10 to the 8. And then I'm going to square that whole thing. And then take the square root of all of it. And now if you punch that into your calculator, you can double-check. With my numbers I got 2.03 times 10 to the minus 6 seconds. So this is something called time dilation. The particle looks like it lives longer as measured by you then it does as measured by itself. Okay, this is the nature of time dilation. And in class, we talked about the weird implications of this, particularly with regards to the twin paradox, right, you put two twins on the Earth, you put one of them in a spaceship, fly them around for a long time, when they come back, they are younger than the other twin. And that's really what happens.