Question: A boat is moving across a river with a speed of 3 m/s. The river current is traveling at 4 m/s. What is the magnitude and angle of the boat’s total velocity? This type of problem usually has a lone rower trying to row his boat directly across the river. However, the current moving pushing downstream causes him to move at an angle! Sometimes they ask you to calculate how far downstream he will be based on the velocity and width of the river, but being able to combine these components into the total velocity vector is very important. In the next five steps we’ll be able to see exactly how to do that.
Step 1: Identify the coordinate axis and draw it on the paper. Free to choose, as always, we will align our coordinate axis with the two component vectors in our problem, i.e. the river current and the rower of the boat. Thus, we have our rather standard cartesian coordinate axis with the x-direction pointing to the right and the y-axis pointing upwards.
Step 2: Identify the components. In this problem we’re even given numerical values: the current is traveling along the x-direction at 4 m/s, and the boat is traveling in the positive y-direction at 3 m/s.
Step 3: Use the Pythagorean Theorem to calculate magnitude. Considering the x and y components to be legs of a right triangle, we’re able to calculate the magnitude of our vector, i.e. the resulting hypotenuse, via the Pythagorean Theorem. In the problem we’re conveniently given some perfect squares that calculate nicely to give us a velocity of 5 m/s.
Step 4: Use arctangent to calculate the angle. Since we’re going to be reconstruction the vector, this will always be the hypotenuse of the triangle. Ergo, we will always be using arctangent to calculate the angle, since the components will be the opposite and adjacent sides. In this example, the x-component can be considered the adjacent side, aka the current of the river and y-component or the boat speed can be considered the opposite side. Arctan(.75) is equal to 0.64 rad. Remember, if you see and angle smaller than 5 degrees, do a little mental math and check if you’re in radians or degrees mode. Your answer will look very silly if you wrote 0.64 degrees!
Step 5: Box your answer, you’re done! So, we have a boat which is ultimately traveling 36.86 degrees w/r/t the direction of the river, at a total speed of 5 m/s, with the help of the stream and our crazy boat rower. This mean’s he’s going about 11 mi/hr across the river. Not bad!
