A boat moving at a constant speed of 10.8 km/h wants to cross a river perpendicularly, whose waters have a constant speed of 1.5 m/s.
a) In what direction should the pilot keep the longitudinal axis of the boat relative to normal to the current?
b) What is the speed of the boat relative to the river bank?

Problem data:

• Speed of the boat relative to the river:    v_b/w = 10.8 km/h;
• Speed of the river relative to the bank:    v_w = 1.5 m/s.

Problem diagram:


Solution

First, we must convert the boat speed given in kilometers per hour (km/h) to meters per second (m/s) used in the International System of Units (SI) a) We see that (Figure 1-A) if \( {\vec v}_w \) is the velocity vector of the river waters, the normal vector to this vector will be given by the vector \( {\vec v}_b \). Relative to the vector \( {\vec v}_b \), the boat must maintain a direction given by the vector \( {\vec v}_{b/w} \) that makes an angle θ with \( {\vec v}_b \) (Figure 1-B). The problem gives us the leg \( {\vec v}_w \) and the hypotenuse \( {\vec v}_{b/w} \) of the right triangle in the figure, the angle θ will be


b) Applying the Pythagorean Theorem to the triangle in Figure 1-B, where \( {\vec v}_{b/w} \) is the hypotenuse, \( {\vec v}_{w} \) and \( {\vec v}_{b} \) are the legs the magnitude will be