I swim at a constant rate. Well, close enough; assume I swim at a constant rate, then we'll adjust that later in this blog. That rate is roughly two meters for every second. Well, ok, that's a little quick for this ol' lady, but those numbers work for us. So pretend.
Let's start by thinking of what a graph would look like as I cruise at my theoretical constant pace of two meters per second back and forth in a pool that is 25 meters long. Let the x axis (independent variable) be time, measured in seconds, and let the y axis represent meters away from the wall, or side, of the pool. Let's start our stopwatch at the exact time when I leave the wall and head to the other side of the pool.
How does the graph appear if I adjust my pace to 1 meter per second? How about to 3 meters per second? 1/2 meter per second?
These kind of "waves" are not quite a sine wave. Wikipedia categorizes the waves as triangular:
http://en.wikipedia.org/wiki/Sawtooth_wave.
But there's more. Lots more.
How would the graph be different if I swam my 2 meters per second pace in a pool that was 50 meters long?
How would I need to swim differently to make the graph look like the other waves that the wiki pictures on the bottom right? Which ones are not possible? Spend some time with, of course, the sine wave and describe how I'd need to swim so that my distance from the wall would be a lovely, smooth sine wave.
So you've already seen that I do some interesting math while I swim. I can solve all the world's problems when I swim, you know. Now what happens if I'm swimming in the pool that's 50 meters by 25 meters. I swim the long way at a constant 2 meters per second and another swimmer decides to swim the short way, perpendicular to my path and swimming a constant 1/2 meter per second. (No kidding, someone actually decided to do this last summer.) I swim in lane one as does he (well, we don't but pretend we do) so we start at the same time and in the same corner. Because we are all thinking perfectly and precisely mathematically, we imagine triangle waves, but let's say we are so far advanced that both the other swimmer and I slow down and speed up appropriately so that the graphs of our distances from the wall with respect to time form perfect sine curves. You've already worked out how we adjust our speeds to make perfect sine curves, of course, because the previous paragraph asked you to do this.
So the bigger question is, assuming we can occupy the same spot simultaneously or that I can duck directly below the surface while maintaining my lovely sine curve pacing, how often do our paths intersect?
For your blog, either show graphs like I've described or take my idea a little further, maybe by answering the questions I pose, OR find your own application of a sine curve. Where do YOU see it in the real world?
In the real world, there are many different ways sine curves can be applied. For example, if you go to the hospital for a fairly serious injury or disease, they will monitor your heartbeat. Because they graph your heartbeat on the amount of beats in a certain period of time, a graph of your heartbeat can be represented, where the x-axis would be the amount of time and the y-axis would be the number of beats. Another application of sine curve could be music, where different frequencies and different amplitudes would get you different graphs, some of which would be sine curves.
ReplyDeleteAs a fellow swimmer, I can really visualize this idea of waves on a graph. One of my favorite things in the world is to bob on the waves at the beach. If you swim at a constant pace forever, the graph would be a slanted, straight line. But here in reality, the graph would almost look like a series of elongated triangles because at every wall the time (x) would keep going and the meters (y) would halt while you flipped. But getting even more real, as you push off the wall your speed would increase, making a curve. To have a sin curve in the pool you'd need to slowly and almost robotically decelerate and accelerate. The sin graph would need to be moved up too, so that when you are at the lowest y point, you are 0 meters from the wall, because that's when your feet are touching the wall. Waves on their own could look very different on a graph depending on the force that drives them. I don't think the square Waves on Wikipedia are possible because it would mean that there would be no motion in the water and then immoderate motion and than none again instantly. This is a very cool way to bring trig to life!
ReplyDeleteGraphing swimming is a lot more complicated than it might seem. The only graph that is somewhat simple is if the swimmer swims at a constant pace throughout the whole time/race. This graph would be a straight line angling at about 45 degrees; or a sawtooth wave. The faster the swimmer, the more compressed horizontally the sawtooth wave becomes. The slower the swimmer, the more horizontally elongated the sawtooth wave becomes. By increasing the distance (50 meter pool), it would take longer to reach the peak of the graph. For this reason the graph would be vertically stretched if the distance was increased.
ReplyDeleteTo make the graph look like a triangle you would have to go a constant pace back and forth. To make the triangle look like a sawtooth wave, you would have to go a constant pace to the other side of the pool. It is impossible to make it look like a square because no human could ever achieve a graph like that. To make it look like a sine curve you would have to increase and decrease your speed at certain intervals of your swim.
In regards to the two swimmers question, the swimmer swimming the long length of the pool will get to the other side twice as fast compared to the swimmer swimming the short distance. The amount of intersections in the graph is one. By the time the slow swimmer gets across the 25m, the fast swimmer will have gone to the other
wall and back. This is the intersection point. The paths of the two swimmers with intersect every time the slow swimmer gets to the other wall. EUS!
So, during normal swimming the graph would be a saw tooth graph. It would rise then drop straight down and then repeat, over and over again. In a longer pool the graph would just be elongated, and it would take longer for it to reach the peak before it dropped down. In order for a swimmer to achieve a sine curb they would have to speed up and slow down, at certain intervals in order to achieve a sine curb. So with the two swimmers the faster one will swim up and back in 50 seconds, where the slower one will have swum up in 50 seconds so they will meet every time the faster swimmer reaches the starting wall.
ReplyDeleteIn order to achieve a perfect sine curve, you would have to start off quickly, then reduce speed. (The sine wave itself does that, starts off climbing relatively quickly, then slows as it reaches its peak, then drops slowly, and finally more quickly.) The points of this particular graph would move closer together if you were to speed up (1 meter per second or 1/2 meter per second) and farther apart if you were to slow down.
ReplyDeleteSome of the graphs shown by Wikipedia are not possible to achieve by swimming. The square one, for example, wouldn't work because you would have to travel a certain distance in an instant. And as cool as that would be, people can't do that (yet.) And as previously stated, to achieve a sine curve, a person would have to swim at a not-constant speed.
Sawtooth waves are also used in music, since they can be used as starting points to create sounds. (I don't personally know much about this, but I have a friend who's really great with synthesizing music electronically.) I think it's really cool that something as common as music can be expressed mathematically.
When swimming at a constant rate of 1 meter per second in a 25 meter long pool, the swimmer creates a saw-tooth wave. If the swimmer swims at a constant rate of 2 meters per second in a longer 50 meter pool, the swimmer creates the same saw-tooth wave; it will just be larger and more elongated. In order to create a perfect sine curve, the swimmer would have to swim very fast for a short period of time and then slow down and once more speed back up. This created the ups and down of a sine curve. It doesn’t sound like the type of swimming a swimmer would want to do in order to win a race. Instead, the swimmer must keep a fast constant rate to win the race. Looking at the other Wikipedia waves, the square wave looks very interesting. When thinking about it, it seems unimaginable because the swimmer would have to literally teleport to a place in the pool and then stop motion all together. That sounds very cool, but takes the point away of swimming entirely. :)
ReplyDelete