Question 3.14: FLYING IN A CROSSWIND An airplane’s compass indicates that i...

IDENTIFY and SET UP:

This problem involves velocities in two dimensions (northward and eastward), so it is a relative velocity problem using vectors. We are given the magnitude and direction of the velocity of the plane (P) relative to the air (A). We are also given the magnitude and direction of the wind velocity, which is the velocity of the air A with respect to the earth (E):

\overrightarrow{\boldsymbol{v}}_{\mathrm{P} / \mathrm{A}}=240 \mathrm{~km} / \mathrm{h}      due north

\overrightarrow{\boldsymbol{v}}_{\mathrm{A} / \mathrm{E}}=100 \mathrm{~km} / \mathrm{h}      due east

We’ll use Eq. (3.35) (\overrightarrow{\boldsymbol{v}}_{P / A}=\overrightarrow{\boldsymbol{v}}_{P / B}+\overrightarrow{\boldsymbol{v}}_{B / A}) to find our target variables: the magnitude and direction of velocity \overrightarrow{\boldsymbol{v}}_{\mathrm{P} / \mathrm{E}} of the plane relative to the earth.

EXECUTE:

From Eq. (3.35) (\overrightarrow{\boldsymbol{v}}_{P / A}=\overrightarrow{\boldsymbol{v}}_{P / B}+\overrightarrow{\boldsymbol{v}}_{B / A}) we have

Figure 3.35 shows that the three relative velocities constitute a right-triangle vector addition; the unknowns are the speed v_{\mathrm{P} / \mathrm{E}} and the angle a. We find

EVALUATE: You can check the results by taking measurements on the scale drawing in Fig. 3.35. The crosswind increases the speed of the airplane relative to the earth, but pushes the airplane off course.