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Physics36. Special RelativityConsequences of Relativity

Okay. I don't know if I want to do 16 here though, so send them a message back saying we might not get to it. Let's take a look at 17 first. So 17 says astronaut Hans leaves the Earth in a spaceship traveling at a speed of 0.28 C relative to the observer, Brian, at rest on Earth. Hans is holding a pencil at an angle of 30 degrees with the direction of travel as seen by Hans. What angle does the pencil make with the direction of travel as seen by Brian on Earth? All right, this sounds like a good difficult problem so let's see if we can visualize what's going on here. Okay, here's Brian standing here on Earth and he's observing Hans flying by in a spaceship, so here is our spaceship, and Hans is standing in there and he's holding a pencil, and that pencil is making an angle of 30 degrees relative to the direction of travel. Okay, and the whole spaceship is moved along at speed V which is 0.28 C. And the question is what angle does the pencil make according to Brian? Okay. So it's kind of a weird question, right? I mean this astronauts flying by and he's holding a pencil up at some angle, it almost seems like this guy should see the pencil at the exact same angle but that's too obvious an answer, right? And in fact we're not going to be suckered into answering A, 30 degrees, so what is really going on here? This is a little bit related to what we said yesterday in class when we talked about length contraction. Okay, length contraction is the following. In the moving frame if you measure L naught by the guy in the spaceship, then L as measured on Earth is going to be shorter. Okay, this number is smaller than one, so this is going to be L not as measured in the spaceship, this is as measured on Earth. Okay, in this case that would be Brian, Brian's the one that stayed on Earth. L naught is the measurement in the spaceship, so that's measured by Hans. Okay, but they're not asking about the length of the pencil, they're asking about the angle that it makes relative to this horizontal axis. Hmm. What can we say? What we said in class yesterday was imagine you have a box and that box has height H naught and length L naught, and now you take that box and you start moving it at some fast speed V. Does the height of the box change? No. The length changes, this length contraction is only in the direction of motion. Okay, so the box shrinks in that direction. So by analogy here, this pencil that is angled looks kind of like a triangle. Here's our pencil. Okay, but there is a horizontal component and there is a vertical component, and that horizontal component, let's call it L naught, and the vertical component, let's call it H naught, and this thing let's call it theta naught. But when it starts moving and Bryan takes a measurement of it, L naught shrinks, H naught stays exactly the same and so in fact the pencil looks like it is more upright, and let's call that angle theta. H naught stays the same but the L shrinks. Alright. What is the tangent of theta naught? The tangent of theta naught is H naught over L naught. What's the tangent of theta? Tangent of theta is H naught over L, but H naught over L is the same as H naught over the square root 1 minus V squared over C squared, all of that times L naught. And H naught over L naught, we know what that is. That's just the tangent of theta naught and so now you can put all this stuff together. And so we have the following. Tangent of theta equals H naught over L naught 1 over square root, 1 minus V squared over C squared, but that is 1 over the square root 1 minus V squared over C squared times the tangent of theta naught. And now you know all of that stuff, you know V, you know theta naught, you can calculate theta. So let's plug in some numbers and see what we get. We've got 1 over the square root 1 minus 0.28. That thing is squared, the C's are of course going to cancel out, and then we have the tangent of 30 degrees and we're going to take the arctangent of that whole thing, and that's going to be our theta. Okay, a few of those answers don't make any sense at all, 30 degrees, that doesn't make any sense because we know the angle is going to change. 29 degrees doesn't make any sense because we know that it's going to get steeper. So the other options are 31 degrees, 33 degrees, and 90 degrees, but 90 degrees, it would be standing straight up and that means that the bottom side of that triangle would have to shrink to 0, which would happen when you're going at the speed of light, so we're not going at the speed of light, so really we're down to two options: 33 degrees or 31 degrees. Did anybody punch it in and get an answer for this? >> (student speaking) 31 degrees. >> 31 degrees.