Sailboat Vector Problem

Question

A boat is sailing across a lake. The boat’s motor can move the boat at a speed of 18 km/hr in still water. Today, there is also a current of 5 km/hr flowing S15° W. Additionally, there is a wind blowing 12 km/hr in the direction of S82° E. If the boat’s destination is due north of where it currently is, what direction should the boat aim? What will be the actual speed of the boat?

Answer 1

We need to find the x and y values of each "opposing" vector before we can add them. Once we have a single vector we can determine the direction the boat must go to cancel them out to acquire a true northern direction.

Looking at the wind speed and direction, we have speed of 5 km/h, 15 degrees west of south (if I'm interpreting this properly). 5 sin 15 = 1.294 (west) and 5 cos 15 = 4.829 (south).

Looking at the current, the speed is 12 km/h, 82 degrees east of south. 12 sin 82 = 11.883 (east) and 12 cos 82 = 1.670 (south).

Combining the east and west components we have -1.294 + 11.883 = 10.589 (east).

Combining the south components we have 1.67 + 4.829 = 6.499 (south).

Since we want to cancel out the east portion of the vector, we must travel in direction θ so that our west vector is 10.589. Our speed is 18km/h so 18 cos θ = 10.589 ⇒ cos θ = 10.589/18 = 0.588. Taking the inverse we get cos^-1(0.588) = 53.965 ≈ 54 degrees

So we must travel at 18km/h in the direction 54 degrees northwest or using the terminology for direction, N46°W