How Should a Riverboat Navigate to Cross a River Efficiently Against a Current?

[mikeyman2010]

Hi, can anyone explain to me in detail the logic/physics to this question? I posted it in general physics, & i guess it's the wrong forum. So i'll post my question here:


A riverboat is trying to go north and cross a horizontal river to the other side. In the river, there is a current blowing to the east. Which path should the boat take so that it will arrive at the other end of the river in the shortest time?

a)North
b)Northeast
c)Northwest

See diagram for more detail:

__________________________________________________

current--->
North?

Northwest? Northeast?


/ \
________________[__ ]<Boat_________________________

Again, please provide a detailed explanation with an answer as well. Thnx!

 

[Hurkyl]

Well, as stated, you're considering 2-D kinematics; you have a boat traveling some direction in two dimensions, you have a desired displacement in the north-south direction, and you have a current in the east-west direction.

The title suggests you have the right idea; you can reduce the problem to one dimension...

 

[Nenad]

It is my guess that he should point the boat straight across to the other shore, or north, since it doesn't matter where he ends up down stream.

 

[BobG]

Northwest.

The current will pull him downstream, so he must aim somewhat upstream to compensate.

Actually, two things happen.

By powering at an angle to the current, some of the boat's velocity compensates for the downstream velocity caused by the current.

Additionally, if he points his boat a little upstream, the current is no longer pushing him straight downstream. By hitting an angled surface, the current is also pushing the boat to the North, across the river. This is how a kayaker is able to ferry across a rapidly moving river in spite of having not nearly enough physical strength to overcome the current - in fact a good kayaker makes this look elegant and effortless (not me, of course - but I have been so fooled by watching a good kayaker that I've misread the current and wound up slaloming through a boulder field, which is pretty exciting). It's also similar to how sailboats tack on upwind courses.

 

[Doc Al]

This question was addressed in this thread: https://www.physicsforums.com/showthread.php?t=34419

 

[mikeyman2010]

Yes, sorry about this, I thought I posted this question in the wrong place, and posted again here. The correct answer is b), pointing it straight across, since we only consider one direction, north, and to get the maximum speed possible would be to point the boat directly across. Anywhere else would be using only a component of the velocity of the boat to get across the river, which would slow the boat down. Did I get that right, Doc Al?

 

[mikeyman2010]

Oh yea, I mean, a), not b)

 

[Doc Al]

You got it!

Did I get that right, Doc Al?

Yep. Go north, young man.

 

[BobG]

After looking at mikeyman's other post and realizing this is for a high school physics class, I agree, North is probably the answer that is being asked for. Still, it's one of those things that stick in one's craw because rivers don't really work this way.

The best angle is always upstream, even if only concerned about how fast you can cross the river with no thought about how far downstream you might drift. If straight upstream were 0 degrees, the best angle of attack (for shortest time, at least) would be somewhere between 45 degrees and 90 degrees (but never exactly 90).

The faster the current and the slower the boat speed, the closer to 45 degrees the boat should be angled. For example, if the current were 10 m/sec (incredibly fast, a class V rapid to be sure) and the boater could only paddle the boat at 1 m/sec, the best angle would be very close to 45 degrees and your boat would travel across the river up to 5 times faster than you could paddle on your own - a substantial gain over relying on your own paddle power (of course, you're traveling downstream almost as fast, which could be a problem).

The faster the boat and the slower the current, the closer to 90 degrees the boat should be angled. If the boat can go 10 m/s (pretty fast for a boat on the river), and the current is only 1 m/s, the boat probably should be angled at least 85 degrees - you start losing more of the cross stream component of your velocity than a weak current can add real quick.

You'd probably need to get into too many details on boat design to actually solve the problem with numbers, but fortunately, the best physics and math is usually solved without a calculator or pen or pencil.

They're just solved by feel - and what a feeling it is.

 

[Hurkyl]

Upon what principle is your hypothesis that there will be a northerly speed boost by paddling upstream based?

 

[BobG]

If you look at the current from the boat's point of reference, with the front of the boat the primary direction, the current's force can be broken into it's forward/backward direction and it's lateral direction. The current will have a backwards component and a lateral component. Because of the hull's design, the force in the front/back direction will be less than what would be mathematically expected. (The boat 1 diagram shows what I mean, although my pathetic artistic abilities prevent me from drawing it to scale).

Rotating back to the river's frame of reference, the boat's lateral direction vector has a component that is lateral to the river. The boat's forward/back vector also has a lateral component, which is pointing away from the direction you want the boat to travel, but it won't be nearly large enough to compensate for the first vector. Both vectors will still have a downstream component. (Diagram 2 shows this, again not to scale)

In practice, you'd need a pretty good paddler to hold a boat steady at an angle between 45 and 90 degrees to the upstream direction. I normally have to keep an angle around 10 to 30 degrees and move pretty slowly cross stream and still drift downstream, since most of my effort is in keeping my boat aligned vs. actual paddling. A powered engine (or a skilled sailor, in the case of sailboats and the wind) could hold its angle much easier.

 

[Nenad]

Northwest.

The current will pull him downstream, so he must aim somewhat upstream to compensate.

Actually, two things happen.

By powering at an angle to the current, some of the boat's velocity compensates for the downstream velocity caused by the current.

Additionally, if he points his boat a little upstream, the current is no longer pushing him straight downstream. By hitting an angled surface, the current is also pushing the boat to the North, across the river. This is how a kayaker is able to ferry across a rapidly moving river in spite of having not nearly enough physical strength to overcome the current - in fact a good kayaker makes this look elegant and effortless (not me, of course - but I have been so fooled by watching a good kayaker that I've misread the current and wound up slaloming through a boulder field, which is pretty exciting). It's also similar to how sailboats tack on upwind courses.

it doesn't ask to get straight across the shore, it asks to get there in the minimum time.

 

[mikeyman2010]

Interesting concept.

 

[Hurkyl]

I don't follow your reasoning at all. The first image does not, at all, represent the decomposition of a force traveling in the direction of the current. Furthermore, this force is only applied during the relatively brief period where the boat is laterally accelerated to the speed of the current. The hull's design only alters the distribution of the force, it doesn't magically increase one of its components.

 

[loseyourname]

Hurkyl, I think his point is that, because of the hull design, the current will push with more force against the bow than against the stern, causing the boat to tilt eastward, so one must row slightly westward to maintain an even keel.

 

[Hurkyl]

Yes, but the question isn't about maintaining an even keel...

 

[Zorodius]

Hurkyl, I think his point is that, because of the hull design, the current will push with more force against the bow than against the stern, causing the boat to tilt eastward, so one must row slightly westward to maintain an even keel.

No, he's not talking about compensating for the current, he's saying the river will exert a northward force on the boat due to some property of the hull's design.

 

[loseyourname]

Yes, but the question isn't about maintaining an even keel...

The point is, the answer is north, but if you actually want to move north, without getting completely caught by the current - making you unable to get across at all - you need to angle slightly westward.

No, he's not talking about compensating for the current, he's saying the river will exert a northward force on the boat due to some property of the hull's design.

This is a separate consideration, but yeah, it should have some effect as well. Think of the way a rudder works. If you angle the rudder so that it is oriented in a northeasterly manner, then the back of your boat will be pushed to the east, so that the front of your boat is now turned west (the point being of course to make a left turn when moving north). Now just take the original boat out of this, then translate east into north, northeasterly into northwesterly (that is, rotate the coordinate frame 90 degrees to the west), and make the rudder the whole boat, you should be able to see the point he is making. Trust me, this does work. Take a kayak or canoe out to a river any time and you can see it for yourself.

I guess the second consideration here is really what Bob G was getting at, after re-reading his posts. The real answer here would be that you need to paddle northward, but angle the boat to the northwest.

 

[loseyourname]

I could demonstrate this really easily with a drawing.

 

[Hurkyl]

The thing is, none of this addresses my primary problem with the northwest answer; once the canoe is up to speed, the current should be irrelevant; it's speed relative to the canoe is zero.

 

[BobG]

Actually, Hurkyl has a point. It's not a sustainable force.

This is a real phenomenon that I've seen others use and that I've used myself(sometimes with complete success - sometimes with at least a little success). But, I think whitewater kayaks are a special case in that:

1) The main channel of mountain rivers is not very wide.

2) Kayaks have no keel.

In other words, there's a large burst of lateral acceleration when you exit an eddy into the main stream and a kayaker can maintain the velocity gained for a longer time - at least long enough to exit out the opposite side of the main channel.

The physics of it are exactly like a rudder if you only look at the forces on the rudder itself (because most boats have keels, you don't see the stern making a dramatic sweep in the direction the rudder's pushing or pulling it - instead you see the bow pushed the opposite direction in an arc).

A kakak has a either a rounded hull or, if a playboat, a flat area on the hull. Neither has a keel. The water slides sideways under the hull very easily, something you notice very quickly when you're first learning to steer. The sides of the boat, being angled more perpendicular to the water, catch water very well.

When you exit an eddy with an angle to the current, you have a large difference between the current speed and the boat speed (the boat's headed upstream at an angle). You get a large burst of acceleration, especially with a slight tilt towards the upstream side to increase the amount of high drag area exposed to the stream (don't tilt so much that the water catches the top of the shell or the boat will flip very rapidly). The current pushes the boat diagonally - with a definite lateral component.

Any velocity gained from that initial burst of acceleration is yours to keep - at least until something decelerates you, such as the resistance of the water. The tilt is reducing the drag in the direction you want to travel since you're reducing the amount of high drag area exposed to the current. And, as long as the boat's downstream speed is less than the current's, force is applied to the boat, even if the force is steadily decreasing as the speed differential steadily decreases. If you maintain a good angle and good tilt all the way through, getting everything the current has to give, the velocity gained from the current can be maintained at least long enough to exit out the opposite side of the main channel.

If you lose that initial velocity for some reason, the only new acceleration you can gain is going to be the difference between current speed and boat speed generated by the paddler - a difference that will never come close to the initial difference when you first exited the eddy. When exiting the eddy the boat actually had an upstream component to its velocity - starting at the speed of the current, the paddler will just have a slower downstream speed than the current. This explains why, when I do get bogged down mid-stream, I can never get that velocity back even if I get my form back. I might be able to paddle hard enough to keep from losing much ground downstream, but I'm having to use such a steep angle that I'm gaining hardly any lateral speed either.

For a normal boat, this isn't going to work very well. The keel makes it hard for the boat to slide sideways and any 'extra' velocity is sucked away as fast as it's gained by increased resistance. Just as a rudder winds up causing the bow to go the opposite direction, some of the extra force may wind up increasing the boat's speed along the keel, but I doubt it would be enough to compensate for the lateral velocity lost by moving at an angle.

And, since even a kayak experiences resistance from the water, this probably wouldn't get you effortlessly across the Mississippi.

As loseyourname mentioned, the current applies more force on the bow than the stern. That's more of a complication than a reason for increased lateral speed. The boat's taper and its angle in the current means the bow is more perpendicular to the current than the sides. The stern's taper mean's less of it is exposed to the current. This is offset a little by the fact that it's also easier for the stern to slide into the protected area behind the body of the boat than outward into a hard current, but the paddler still has to actively compensate for the torque to maintain a good angle to the stream or lose his momentum.

 

[Hurkyl]

I'll buy using a current differential to cause a deflection. I had no way of knowing if it was reasonable or not to use variations.

 

[mikeyman2010]

Er, this is too deep for me... lol. Do I need to worry about this stuff in high school, cus I have NO idea what you guys are talking about...

 

[HallsofIvy]

I'm now kind of lost as to what question is being discussed! Getting back to the original problem, if the problem is just to get across the river as fast as possible, without regard for where on the opposite bank you land, then you should aim directly across the river without regard for the current.

The whole point of the "vector" method is that the components are independent.