Question

You are traveling on an airplane. The velocity of the plane with respect to the air is 140 m/s due east. The velocity of the air with respect to the ground is 31 m/s at an angle of 30° west of due north. 1) What is the speed of the plane with respect to the ground?

Answer 1

Answer:

127.36 m/s

Explanation:

velocity of plane with respect to air = 140 m/s due east

velocity of air with respect to ground = 31 m/s 30° west of north

Write the velocities in the vector forms

$\overrightarrow{V_{p/a}}=140\widehat{i}$

$\overrightarrow{V_{a/g}}=31 \left ( -Sin30 \widehat{i}+Cos30\widehat{j} \right )$

$\overrightarrow{V_{a/g}}= -15.5 \widehat{i}+26.85\widehat{j}$

Let velocity of plane with respect to ground is given by vp/g

According to the formula of relative velocities

$\overrightarrow{V_{p/a}}=\overrightarrow{V_{p/g}}-\overrightarrow{V_{a/g}}$

$\overrightarrow{V_{p/g}}=\overrightarrow{V_{p/a}}+\overrightarrow{V_{a/g}}$

$\overrightarrow{V_{p/g}}= \left ( 140-15.5 \right )\widehat{i}+26.85\widehat{j}$

$\overrightarrow{V_{p/g}}= \left ( 124.5 \right )\widehat{i}+26.85\widehat{j}$

The magnitude of the velocity of plane with respect to the ground is given by

$V_{p/g} = \sqrt{124.5^{2}+26.85^{2}}=127.36 m/s$

Thus, the velocity of plane with respect to the ground is given by 127.36 m/s.