字幕表 動画を再生する 英語字幕をプリント This video contains the answers to my four revolutionary riddles, so if you haven't seen the riddles yet, you should probably watch them before you watch the answers. It's OK; I'll wait. Just click this card up here. [Ticking clock sound] Now, when I filmed the riddles, I also filmed the solutions at the same time, but that was before I received your 15,000 comments and dozens of video responses, so I'm re-shooting parts of this solutions video to incorporate the results I saw in the comments and to use some of your video responses to help explain the solutions. Let's get to it! OK, by looking at the comments, for #1, over 15% of you said "a cylinder containing sand", Now, this contains some powder. [Chuckles] See how it rolls... It, uhh, it doesn't seem to roll... ...very far before it stops, and then it won't roll again because I think the sand just kind of levels off in there. Maybe you were thinking a bigger kind of sand, like these small gravely stones. Let's try that. That actually rolls pretty well. Nearly 25% of you said, "a cylinder half-full of water," so let's try that. This rolls very well, so it's not water... and nearly 45% of you said, "a cylinder half-full of a viscous liquid." Here, I have a half-full container of honey, so let's see how it rolls... That's not bad; it's rolling and it's stopping, and it's rolling some more... This is a pretty good guess, and I think the behavior is not exactly the same as the mystery cylinder, but it definitely is similar, and that's no coincidence. The mystery cylinder actually contains honey... and ping pong balls. There are two ping pong balls submerged in this honey. So if I place that on the ramp, the center of gravity is not above the point of contact with the ramp, and so it rolls forward, but now, because those ping pong balls are in the front, they change the center of gravity, and so it's exactly over the point of contact, and so it stops briefly, but then, as the viscosity of the honey allows those ping pong balls to move up, the center of gravity shifts forwards again, allowing this little container to roll. So that is the trick of the mystery cylinder. Pretty easy if you want to try it out at home. Now, I challenged you to run two laps of this track, where the first lap, you could go as slowly as you like, but the second lap, you had to go much faster, such that your total average speed was twice the speed of your first lap. Now, when I was first asked this question by Simon Pampena, it took me a long time and scribbling on paper, and just something didn't seem to work out, and that's because... you can't actually do this. It's impossible. I mean, you might think I could run 3V₁ for my 2nd lap, and that would mean my total average speed is 2V₁. The problem is you can't just add the two speeds together and divide by two, because you spent much more time in your first lap, so that speed is weighted more heavily into the average, so you'd have to run, well, impossibly fast. Let me explain. The velocity of the first lap was the distance around the track divided by t₁, the time it took you. Now, if you want your total average speed to be 2V₁, well then it needs to be 2d ÷ t₁. You need to run twice the distance in the same amount of time it took you to run the first lap, but you've already run that first lap, and so you have no time remaining to run the second lap! Even if you went the speed of light, you would not be able to increase your total average speed up to twice the velocity of your first lap. It is just mathematically impossible! So this may seem like a bit of a trick question, but the point to me is how doable it sounds, how it seems like something you should be able to do, but you can't. It's actually impossible. Riddle #4, the question about the train, was actually answered pretty well, with most people mentioning something to do with the wheels, but of course, that makes sense, in a series of riddles which are about rotation, rotational motion. Some people though did point out that maybe it was the steam that was going backwards, or maybe air molecules in the train, and that is actually a pretty clever point, however I wouldn't really consider the air in the train part of the train, so indeed, the part of the train that is moving backwards is the flange part of the wheel, which is below the rail. That is the part of the train which is moving backwards. To understand why, you just think about a spinning wheel. The top of the wheel is moving forwards at speed 2V, and the bottom of the wheel is not moving forward at all; it is stationary with respect to the track, and that is what we call "rolling without slipping," and that's how most wheels work. At least, that's how they're designed to work. Now, in the case of trains, they have to have flanges so the train doesn't fall off its rails, but of course, when these pieces come around during the rotation of the wheel, they actually extend beyond the rail, and therefore, they are going backwards with respect to the ground, so the part of the train that's moving backwards is always changing, but it's always that part of the flange, that part that extends beyond the wheel that is below the level of the track. So what happens when you pull the bottom pedal of a bike backwards? Well, about 45% of you thought that the bike would move backwards, about a quarter said it would move forwards, and a quarter said the bike wouldn't move at all, and 5% said it depends on something... So let's give it a shot and see what happens. I'm going to pull backwards on the bike pedal in 3, 2, 1... Woah! The bike did indeed move back, and for virtually all bikes, this is what you will find, but the explanation is not just as simple as "well, the net force on the bike is back, so therefore, it has to accelerate backwards," and to prove that that logic doesn't work, well, just have a look at this video by George Hart. [George] Watch this. Again, I pull the same pedal backward, but now... the bike moves forward! [Derek] I'll put a link to the full video here and a link to his website in the description. So the reason the bike moves backwards is because of the way these gears are set up, the diameter of the tire, and also, the distance from this crank to the pedal itself. Because as a bike moves forwards, the pedal, even when you're pushing back on it, never actually moves backwards with respect to the ground; it's always moving forwards. [George] So if you drag a string behind the pedal of a bike moving forward, the string is always moving forward. Now just play that movie backward in your mind, and it may be clear how pulling the string backward could make the bike move backward; They move forward together, so they move backward together. [Derek] Another way to think about this is to consider the path traced out by the pedal as the bike moves forward. This is called the trochoid. For all ordinary bikes, the pedals are moving much slower than the tires, so the pedal is always going forward with respect to the ground, but George modified his bike so the ratio of the pedal to the wheel radius was greater than the ratio of the front sprocket to the back sprocket, and this ultra-low gear changes the trochoid so the pedal DOES actually go backwards with respect to the ground as the bike goes forwards, and that's why he could pull back on the pedal and make the bike go forwards. This is the same reason why if you were to pull backwards on the flange of a train wheel, you could actually get the train to move forwards, if you pulled backwards with enough force. For all normal bicycles, pulling back like this on the bottom pedal will cause the bicycle to move backwards, but, depending on the gear ratio, you can get the bike to move forwards, so it was those people who said it depends on the ratio of these gears and the size of this crank to the radius of the back wheel that were actually the most correct.