A river is flowing due east at 7 mi/h. A man heads his motorboat in the direction N 30° E in the river. The speed of the motorboat relative to the water is 10 mi/h.

Question

A river is flowing due east at 7 mi/h. A man heads his motorboat in the direction N 30° E in the river. The speed of the motorboat relative to the water is 10 mi/h. (Let east-west be represented by the first component, and north-south be represented by the second component.)

(a) Express the true velocity of the motorboat as a vector.

(b) Find the true speed and direction of the motorboat. (Round your answers to one decimal place.)

Answer 1

a) Let v be the true velocity of the motorboat, v_r - velocity of the river, and v_m - velocity of the boat with respect to the water.

Then we can write that the true velocity of the boat with respect to the bank is v = v_r + v_m.

b) To find the true speed v we can use cosine theorem for the triangle formed by v, v_r and v_m.

We see that angle between v_r and v_m is 120°.

We will write

v² = 7² + 10² - 2·7·10·cos 120°.

Hence the true speed v = 14.8 mi/h

Angle between the vertical (North direction) and true velocity of the motorboat

Θ = α + β.

Here α = 30° - angle between vertical and v_m; β is the angle between v_m and v.

To find value of β, we will use the sin theorem for the triangle we used before.

Writing

14.8 / sin 120° = 7 / sinβ

From this equation β = 24°

And then Θ = 30° + 24° = 54°

Answer: 14.8 mi/h, 54°