The international space station travels in orbit at a speed of 7.67 km/s. If an astronaut and his brother start a stop watch at the same time, on Earth, and then the astronaut spends 6 months on the space station, what is the difference in time on their stopwatches when the astronaut returns to Earth? Note that 6 months is about 1.577 x 10 7 s, and c = 3 x 10 8 m/s.

Hey, guys. In this problem, we're going to measure the effects of time dilation for an astronaut on the International Space Station. So So let's say that an astronaut is going to go into orbit in the space station for 6 months, and he and his brother, before he leaves, they synchronize their watches. Okay. So their watches are perfectly synchronized, then he spends 6 months on the International Space Station. When he gets back, wants to compare the watches. How different is the time that they're measuring going to be? Okay. So first of all, if this is the surface of the earth and this is the International Space Station whizzing by it at 7.67 kilometers per second, we are going to call this frame the proper frame, frame because it is the one that is moving relative to the lab frame, the frame on earth. Okay. So the the proper time is going to be the time measured by the astronaut in the, space station, and it's going dilated time is going to be the time measured on Earth. Right? Because the astronaut knows I have to be up there for 6 months, and then I get to come home. The guy on there for 6 months and then I get to come home. The guy on earth is measuring a time near 6 months, but technically not the same. So when the astronaut returns, they can measure the difference. Okay? Now as I said in the previous problem, when we were talking about time dilation for the first time, that was for the escape velocity of earth which was 11.2 kilometers per second, and that basically has no effect on time dilation. This is traveling even more slowly than that, so it's gonna have an almost immeasurable time, of an almost immeasurable effect on the measuring of time. But let's see what it is anyway. So first of all, the proper time is going to be the time measured by the guy on the space station, the, astronaut. And that's going to be 6 months, which is 1.577 times 10 to the 7 seconds. The dilated time is going to be measured by his brother on earth, and that's going to be gamma times the proper time. Right? Which is going to be 1 over the square root of 1 minus u squared over c squared times delta t naught. Okay? Let me just look at gamma by itself really quickly. 1 over the square root of 1 minus well, what is u? U is 7.6 7 times 10 to the 3, divided by the speed of light, which is 3 times 10 to the 8, squared. And then all of this square rooted. Okay? This is so so so so so small. If you actually measure the interior of the square root, you're going to get, what was it, 0.99. It's 1, 2, 3, 4, 5, 6, 7, 8, 9. 1, 2, 3, 4, 5, 6, 7, 8, 9, and then 3. If you then take the square root of that, you're going to get 0 points and it's still yep. 1, 2, 3, 4, 5, 6, 7, 8, 9, 7. And then if you then take the reciprocal of that, right, because it's one over this, what you are going to get is 2, 3, 4, 5, 6, 7, 8, 9. 1.1, 2, 3, 4, 5, 6, 7, 8, 9, 33. Okay? So that is how little an effect this has. You need 9 zeros before you see anything different than just 1. Okay? So let me minimize myself here really quickly. This is going to be that 1.123 3456933 times 1.577 times 10 to the 7 seconds. And plugging this into your calculator, you're going to get 15,770, sorry 15,770,000. So this is exactly what we started with. Everything after this decimal place is gonna be the difference. 0.0052 seconds. That is how little the difference in the time measured by the astronaut's brother on earth is from the time measured by the astronaut in the space station. Right? It is a tiny, tiny fraction of a second. Over 6 months, you're talking about 0.5 percent of one second. Right? 5 milliseconds. Okay? Just as a heads up, if you guys try to look this up, it's somewhere on Wikipedia. It's going to be a slightly different answer because, you also have time dilation due to general relativity, which needs to be taken into account. But this is only looking at the time dilation effects of special relativity. So don't necessarily expect to find this exact number just in case you're like, hey, Doug. I think you're wrong. I'm gonna look this up. It's not that number. Okay? Either way, the number is basically immeasurable. It's so so so so small, the difference between the time measured on earth, the dilated time, and the time measured in, the space station. Alright, guys. Thanks so much for watching, and I'll see you guys in another video.

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