How fast will a boat that normally travels 10km/h in still water be moving with respect to land if it sails directly across (perpendicularly) a stream that’s flows at 10km/h ?

If a boat that has a speed of 10km/h in still water is crossing a stream that flows at 10km/h, the actual speed of the boat with respect to land will be the result of the vector combination of the boat's speed and the stream's speed.

Since the boat is moving perpendicularly across the stream, the motion of the boat and the flow of the stream create a right-angled triangle, where the boat's velocity vector is one side, the stream's velocity vector is another side, and the resultant velocity vector (the boat's actual path with respect to the land) is the hypotenuse.

To calculate the boat's actual speed with respect to the land, we need to use the Pythagorean theorem because we are dealing with a right-angled triangle.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):

c² = a² + b²

Where a is the boat's speed in still water (10km/h), b is the stream's speed (10km/h), and c is the boat's resulting speed across the stream.

Substituting the known values:

c² = 10² + 10² c² = 100 + 100 c² = 200

Now, take the square root of both sides:

c = √200 c ≈ 14.14 km/h

Therefore, the boat's speed with respect to the land while sailing perpendicularly across a stream flowing at 10km/h would be approximately 14.14 km/h.