Chapter 1, Problem 26Q

To determine

To rank the situation of the person on the bases of velocity and direction of motion of person and the train.

Explanation of Solution

Introduction:

An observed velocity or relative velocity depend on the velocity and direction of the person and train.

Situation a:

The person is moving in west direction with speed $2{\text{ms}}^{\text{-1}}$ and the train is also moving in the east direction with the speed $20{\text{ms}}^{\text{-1}}$. Both are moving in opposite direction therefore the resultant observed velocity is,

$\begin{array}{c}{v}_a=20{\text{ms}}^{\text{-1}}-2{\text{ms}}^{\text{-1}}\\ =18{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is in east direction so it is positive.

$\overrightarrow{v_a}=18{\text{ms}}^{\text{-1}}$

Situation b:

The person is moving in east direction with speed $6{\text{ms}}^{\text{-1}}$ and the train is also moving in the west direction with the speed $20{\text{ms}}^{\text{-1}}$. Both are moving in opposite direction therefore the resultant observed velocity is,

$\begin{array}{c}{v}_b=20{\text{ms}}^{\text{-1}}-6{\text{ms}}^{\text{-1}}\\ =14{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is in west direction so it is negative.

$\overrightarrow{v_b}=-14{\text{ms}}^{\text{-1}}$

Situation c:

The person is moving in west direction with speed $4{\text{ms}}^{\text{-1}}$ and the train is also moving in the west direction with the speed $30{\text{ms}}^{\text{-1}}$. Both are moving in same direction therefore the resultant observed velocity is,

$\begin{array}{c}{v}_c=4{\text{ms}}^{\text{-1}}+30{\text{ms}}^{\text{-1}}\\ =34{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is in west direction so it is negative.

$\overrightarrow{v_c}=-34{\text{ms}}^{\text{-1}}$

Situation d:

The person is moving in west direction with speed $10{\text{ms}}^{\text{-1}}$ and the train is also moving in the east direction with the speed $10{\text{ms}}^{\text{-1}}$. Both are moving in opposite direction with the same velocity therefore the resultant observed velocity is,

$\begin{array}{c}{v}_d=10{\text{ms}}^{\text{-1}}-10{\text{ms}}^{\text{-1}}\\ =0{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is zero.

Situation e:

The person is moving in east direction with speed $2{\text{ms}}^{\text{-1}}$ and the train is also moving in the west direction with the speed $20{\text{ms}}^{\text{-1}}$. Both are moving in opposite direction therefore the resultant observed velocity is,

$\begin{array}{c}{v}_e=20{\text{ms}}^{\text{-1}}-2{\text{ms}}^{\text{-1}}\\ =18{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is in west direction so it is negative.

$\overrightarrow{v_e}=-18{\text{ms}}^{\text{-1}}$

Situation f:

The person is moving in east direction with speed $4{\text{ms}}^{\text{-1}}$ and the train is also moving in the east direction with the speed $25{\text{ms}}^{\text{-1}}$. Both are moving in same direction therefore the resultant observed velocity is,

$\begin{array}{c}{v}_f=4{\text{ms}}^{\text{-1}}+25{\text{ms}}^{\text{-1}}\\ =29{\text{ms}}^{\text{-1}}\end{array}$

Resultant velocity is in east direction so it is positive.

$\overrightarrow{v_f}=+29{\text{ms}}^{\text{-1}}$

Thus, velocity of the walker on the bases of magnitude is ranked as,

$v_c>v_f>v_a=v_e>v_b>v_d$

Here, runner a and e are walking with the same observed velocity but in opposite direction. The velocity of the walk c is moving in negative direction with the maximum velocity. The walk d seem to be at rest.

Conclusion:

Thus, the on the basis of relative observed velocity is ranked as,

$v_c>v_f>v_a=v_e>v_b>v_d$.