A motorboat can move with a maximum speed of 10 m/s relative to the water. A current isflowing at 5 m/s [E]. What course (direction) should the boat take in order to cross a 400 mwide river and land at a point directly opposite to the starting point? How long will the triptake? (Answer: [N30oW]. 46 s)

To solve this problem, we need to consider the velocity of the boat relative to the water and the velocity of the current.

Step 1: Determine the velocity of the boat relative to the ground The boat can move with a maximum speed of 10 m/s relative to the water. However, the current is also moving at 5 m/s to the east. This means that if the boat tries to move directly north, it will actually be pushed off course to the east.

Step 2: Determine the direction the boat should head To counteract this, the boat needs to head slightly west of north. This can be calculated using trigonometry. The boat's velocity forms the hypotenuse of a right triangle, with the current's velocity and the desired velocity (directly across the river) as the other two sides. Using the Pythagorean theorem, we can find that the boat should head √(10^2 - 5^2) = √75 m/s to the north.

Step 3: Calculate the angle The angle can be calculated using the inverse tangent function (arctan). The opposite side is the current's velocity (5 m/s) and the adjacent side is the desired velocity (√75 m/s). So, the angle is arctan(5/√75) = 30 degrees. Since the boat needs to head west of north, the final direction is N30°W.

Step 4: Calculate the time it takes to cross the river The width of the river is 400 m and the boat is moving √75 m/s to the north. So, the time it takes to cross the river is 400/√75 = 46 seconds.

So, the boat should head N30°W and it will take 46 seconds to cross the river.