Question

A motorboat heads due east at 12 m/s across a river that flows toward the south at a speed of 3.5 m/s. What is the resultant velocity relative to an observer on the shore?

Answer 1

The resultant velocity of the boat relative to an observer on the shore is 12.5 m/s at an angle of 16.7 degrees south of east.

Step-by-step explanation:

To find the resultant velocity of the boat relative to an observer on the shore, we need to use vector addition. We have the velocity of the boat going east, which is 12 m/s, and the velocity of the river flowing south, which is 3.5 m/s. To add these velocities, we create a right triangle with the boat's velocity as the horizontal leg and the river's velocity as the vertical leg. The resultant velocity is the hypotenuse of the triangle. Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:

R = sqrt((12^2) + (3.5^2)) = sqrt(144 + 12.25) = sqrt(156.25) = 12.5 m/s

The direction of the resultant velocity can be determined using trigonometry. The angle θ can be found using the inverse tangent (tan−1) function:

θ = tan−1(3.5/12) = tan−1(0.2917) ≈ 16.7 degrees south of east