I started this post yesterday thinking this would be a simple, interesting math problem. Turns out I was wrong. Here it is, a day late and one equation short. This past Sunday I participated in the Great Chesapeake Bay Swim
. It's a swim that starts on the western shore of the bay at Sandy Point State Park
and ends 4.4 miles later on the eastern shore at Hemingway's Marina
. The rules are simple, get in the water and swim to the other side while staying between the spans of the Chesapeake Bay Bridge
. The Bay Swim is an Annapolis tradition and has been going on for 22 years. The currents make it a very challenging course, meaning that by the time I'd finished I'd most likely swum farther than the 4.4 miles of the bay. I also had to constantly change my swimming angle with respect to the shore to make sure I didn't get sucked out to sea. After recovering from the "race" I wanted to go back and find out 3 things; the angle at which I should have been swimming to go forward but not get swept out, how far I actually swam, and if I was doing this perfectly, how long should it have taken me. So here are my calculations. Hopefully this will be useful to someone next year!
First a little background on the bay and swimming in it. Swimmers go from west to east. The most significant challenge is the currents caused by the tides
. The tide changes quickly and can be very strong. It can take swimmers anywhere from 1:45 to 4 hours to reach the other side of the bay and the tides go from low to high in as little time as 7 hours
. The race is usually timed so that the majority of racers will start out with a bit of tidal current pushing them north, swim the majority of the race during the wonderful still water of slack tide (when the tide is changing directions) and finish as the tide starts to go out, creating a current pulling swimmers south.
This year, however, we were not so lucky. The race started during slack tide and we felt an increasing pull south as we swam across. As we were getting more tired, the current was getting stronger. When one is bobbing in the bay for an extended period of time, one has lots of time to think. I remembered back to an intro physics problem of a boat in a current trying to get to the other side. The boat had to point at the correct angle to get across successfully. I spent a long time trying to experimentally figure out the correct angle in the bay. Really, I ended up feeling more like this guy
Here is a simplified diagram of the situation:
I made some pretty big assumptions here. First, all speeds are in mph, I had to convert the current speed from knots to mph. I looked at the tidal currents for that day and they started at roughly zero and after about 3 hrs were at 1 mph and they increased pretty linearly. I'm also assuming that I kept moving forward at 2 mph throughout the whole thing. This is a good estimate at my speed in still, open water.
The first thing I wanted to know was the optimum angle at which I should swim. This would be super easy if the current were constant, but isn't too difficult to do with a changing current. Since I want the direction I'm headed to be perpendicular to the direction of the current, I can move the arrows around to be a right triangle.
If I want to successfully finish the race I need to make sure I am going north at least as fast as the current is pushing me south. If I want to finish as fast as possible, I don't want to be going farther north than I have to. This means that I want my northern component to exactly equal the current.
And now I want to see how my angle would change with time so I have to solve for:
A graph of that can give a better idea of which way I should be heading(y is theta in degrees and x is time):
When I start out I should be heading directly towards the other shore but as I swim along, I'll gradually be turning north by about 10 degrees an hour. That may not seem too significant, but every 10 degrees I turn north means a little bit less forward motion towards the other shore and it's a bit more likely I'll get swept out to sea if I stop to fix my goggles. By about 3 hours in, only 2/3 of my forward motion is getting me to safety and 1/3 is going to fight the current.
What is extremely interesting is that by the end of the swim (around 3 hrs) I had decided to swim at about 45 degrees towards the north which means I was going more slowly than I needed to be. I am going to remember that for next time.
The other two questions, how fast could I have gone and how far did I actually swim, are much, much more difficult. In fact, I still haven't been able to solve them. I'll try this weekend and hopefully put up an addendum to this post next week.
Why were those two questions so hard to answer? It seems like it should be pretty obvious to figure out how far you've gone or how long it should take you to go some distance as long as you know the velocity. In this case R(t) is my velocity towards the shore and once I set foot on the beach at Hemingway's Marina I will have gone 4.4 miles. Distance traveled is the velocity times the time you've been moving. The problem is that my velocity is changing with time. This means I have to add up my velocities at every second in time and multiply by that small chunk of time.
Doing that is called integrating and it is extremely useful but can also be rather complicated. I can find R(t) now that I know the optimum angle of attack, but it isn't easy to integrate. However, I can graph it easily enough and show how my velocity changes over time. It turns out that if current kept increasing in the same way and the bay was longer than 4.4 miles, after 6 hours of swimming I would no longer be able to keep it up (y is velocity east and x is time).
I start out going 2 mph towards the opposite shore and then after about 3 hours I'm down to 1.7 mph. It drops off much more quickly after that.
Hopefully I can do some integrals this weekend and figure out how far I swam and how long it should have taken me. I started doing this calculation to get a better sense of why my time was so much slower than what I thought it should be. What I've managed to do in the process is convince myself that I did indeed swim quite slowly, I did a terrible job of heading in the right direction and I'm not as good at doing physics problems as I used to be. But, I finished the bay (with no wetsuit) and I did most of this calculation so I guess I also learned that I have more persistance than I thought. Though I think my boss summed it up best by saying "It shouldn't take you longer to do the problem than it took you to swim the bay!"