Can you sail up/down a moving river on a windless day? It's not possible to sail into the wind directly, but only at an angle. It can be shown (it may help for below) by considering the pressure forces on a flat sail and flat keel that sailing upwind is only possible because of the angle between the sail and keel.
Suppose a sailboat is on a long, wide, straight river with a current of $0.5\,\text{m/s}$ on a windless day (there's no wind with respect to the river bank).


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*Can the sailboat travel faster downstream than $0.5\,\text{m/s}$ by raising its sails and sailing?

*Is there any way the sailboat can sail upstream on the slow-moving river on this windless day?

The source of this question is a bathroom physics problem posted above university physics department toilets near grad offices. Problems are submitted by department members. To the best of my knowledge, none of these were ever homework questions. This question is minimally modified from a problem that was provided by Matt Kleban.
 A: The answer to 1 is yes. It is easiest to analyze all this in the frame of the water.
In the frame of the water there is a $0.5\frac{\mathrm{m}}{\mathrm{s}}$ wind directed upriver. Sailboats are able to sail into the wind, albeit at an angle. This is called sailing close-hauled, or beating. In the frame of the water the boat will be moving into the wind --- albeit at an angle --- and hence in the frame of the riverbank the sailboat will be moving downriver faster than $0.5\frac{\mathrm{m}}{\mathrm{s}}.$ You can see how close-hauled sailing works in this sketch

The sail acts like an airplane wind. As the wind passes the sail the wind is redirected, creating both lift and drag. If the lift is large enough and the angles are right the net force can have a component that pushes the boat forward. This will accelerate the boat into the wind at an angle until the drag from the water balances the driving force from the sail.
The answer to 2 is also yes. Sailing upriver in the riverbank frame means sailing downwind faster than the wind in the frame of the river. Modern sailboats are actually able to move downwind faster than the wind by sailing at an angle to the wind. See "High-performance sailing", Wikipedia. So it is possible (in theory) to sail upriver in this case.
Here is a sketch demonstrating how this works.

The boat sails downwind at an angle to the wind, called "broad reach". As it picks up speed the wind in the frame of the boat changes direction. If the downwind component of the velocity is greater than the speed of the wind then in the boat frame it looks like the boat is sailing into the wind. Since we already know that boats can sail into the wind, the boat can maintain its downwind velocity.
You may wonder how the boat can get going fast enough to do this in the first place. To understand how this could happen, first imagine sailing sideways on the river from bank to bank. The only limit on how fast you can go in this direction is the drag from the water and how well your sails are designed; the $0.5\frac{\mathrm{m}}{\mathrm{s}}$ crosswind can accelerate you to arbitrary speeds. First attain a very high speed this way.
Next, turn your boat upriver. If you angle your boat right and you obtained a high enough speed, the upriver component of your velocity can be arbitrarily greater than $0.5\frac{\mathrm{m}}{\mathrm{s}}$ and you so you will be moving upriver in the riverbank frame at arbitrarily high speeds. You can then set your sails for the apparent wind to maintain your faster-than-downwind motion.
A: Although it doesn't look like traditional sails, it's possible to go faster than the wind both upwind and downwind with a combination of air-propellors and drive-propellors.  It's even been done experimentally:

Last time around, the Blackbird cart raced downwind at 2.86 times the speed of the wind. Earlier this month, Rick Cavallaro and the Blackbird team braved 104 degree heat at the New Jerusalem airport in Tracy, California, clocking in a top speed 2.01 times faster than the wind speed when headed upwind


(image from the linked Wired article)

Where the prospect of traveling downwind faster than the wind once inspired thousands of internet arguments and heated debates in physics classrooms, an upwind sail just isn't as provocative. In fact, there's already a racing series in the Netherlands devoted to upwind land surfing.

The only remaining issue is whether you can have efficient enough water propulsion to do the same.

A: Yes. If the wind is moving fast enough, and moving sideways across the river. The wind will hit the bottom of the sail if it is at a slight angle, and therefore push it up stream. The speed of wind required depends on a lot of factors having to do with the boat, but it would be possible with high enough winds perpendicular to the riverbank. Similarly it could go faster downstream if you rotated the sail 180 degrees from where it travels upstream.
