36. Special Relativity

Consequences of Relativity

# Time Dilation for a Muon from the Atmosphere

Patrick Ford

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Hey, guys, let's see this problem. Okay? We have mu ons which are very, very tiny charged particles similar to an electron, but heavier. They're admitted from very, very high up in the atmosphere when high energy particles from the sun collides with the atmosphere. So if here's the earth's surface, there's a bunch of right atmosphere, just a Ramallah cules. High energy particles coming from the sun collide with atoms inside the atmosphere and produce Mulan's. They're given by the Greek letter mute those Mulan's travel at 90% the speed of light and, as measured in a lab right, measured with respect to the mu on, it's going to decay a 2.2 microseconds right in their rest frame. So that is the proper time, because we're talking about Mulan's moving at 0.9 the speed of light. So this is a mu on, and this is measured in the rest frame. So here's some dude at rest with respect to the surface, watching Um, yuan fly by at 90% the speed of light but in the yuan's own frame as prime right where that frame was moving at 90% speed, the light, the Muan is static. Okay. The muan has no speed. This is technically the prime. It has no speed and it's going to decay in an amount of time of 2.2 microseconds. This time is the proper time. Okay, because the event that we're interested in is the decay of them Yuan. Okay, The moving clock measures time Mawr Slowly. So in the lab frame. When this guy sees them, yuan fly by. He's going to see them. You on live for a longer time because he will be measuring the dilated time. Okay, Now, what exactly is that amount of time? Well, Delta T Prime is going to be gamma times Delta t not. Which is going to be one over the square root of one minus. You squared over C squared times. Delta t not. Okay. Now, most of these problems the speed you is going to be given in terms of the speed of lights. If we look at this term right here, you squared. Divided by C squared is the same as you divided by C squared. So if we plug in 0.9 c divided by C, you'll see that those speeds of light cancel. So this is just gonna be one minus 10.9 square. And this is typically how these problems they're going to be given every now and then instead of giving it something like 0.9 c, they'll say, like three times 10 to the 6 m per second. But most of these problems, they're going to be given in terms of the speed of light because it just makes the calculation more easy. It just makes it easier. Okay, so gamma. The Lawrence factor. If you plug this into your calculator, you're going to get about 2.29 And this is times 2.2 microseconds, which remember 2.2 microseconds, is the proper time. So the Lawrence Factor says that the dilated time is 2. times larger than the proper time, and this is going to be about five microseconds. So the time that an observer on Earth right in the lab frame measures for them you want to decay is five microseconds, whereas the mu on in its rest frame decays in 2. microseconds. Okay, so this is a perfect example of what's the proper frame? What's the lab frame. What's the proper time? What's the dilated time? The event that we're interested in is the decaying of them yuan. And when we measure that time at rest with respect to the mu on, that's the proper time. When we're watching them, yuan zip. By the time that we're going to measure it, taking to decay is going to be the dilated time. Because we're measuring in the lab frame, not in the proper frame. Okay, Alright, guys, that perhaps with this problem, thanks so much for watching.

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