Question

A truck travelling due north at $20\text{m/s}$ turns west and travels at the same speed. The change in its velocity is

• A. $40\text{m/s}$ northwest
• B. $20\sqrt{2}\text{m/s}$ northwest
• C. $40\text{m/s}$ southwest
• D. $20\sqrt{2}\text{m/s}$ southwest

Answer 1

Answer: (D)

Step 1, Given data

The truck is travelling with the speed =$20m{s}^{-1}$ due north

The truck turns in west and travel with the speed= $20m{s}^{-1}$

Given situation can be drawn as

Step 2: Concept and formula used:

1. The change in velocity of a body is given by the difference between the final velocity and initial velocity of the body.
2. The unit vector along the positive direction of the x-axis is given by $\hat{i}$ and along the negative x direction is given by $-\hat{i}$ whereas the unit vector along the positive direction of the y-axis is given by $\hat{j}$ and along the negative x direction is given by $-\hat{j}$ to represent the direction of a vector along the axis.
3. If the velocity is given in the vector form that is;

$v=v_x\hat{i}+v\widehat{{}_{y}j}$

The magnitude of the resultant is given by;

$\left|\overrightarrow{v}\right|=\sqrt{(v{{}_x}^2+v_y^2)}$

Step 3: Calculation of the change in its velocity:

The velocity of the truck in the north direction can be represented in the vector form as;

${\overrightarrow{v}}_1$ = $20\hat{j}$

The velocity of the truck in the west direction can be represented in the vector form as;

$\overrightarrow{v_2}$ =$-20\hat{i}$

Thus,

Change in velocity($\Delta \overrightarrow{v}$)= $\overrightarrow{v_2}-\overrightarrow{v_1}$

$\Delta \overrightarrow{v}=-20\hat{i}-\left(20\hat{j}\right)$

$\Delta \overrightarrow{v}=-20\hat{i}-20\hat{j}$

$\Delta \overrightarrow{v}=-20\left(\widehat{i+\hat{j}}\right)$

Now,

The magnitude of the change in velocity will be;

$\left|\overrightarrow{\Delta v}\right|=\sqrt{{(-20m{s}^{-1})}^2+{(-20m{s}^{-1})}^2}$

$\left|\overrightarrow{\Delta v}\right|=\sqrt{400+400}$

$\left|\overrightarrow{\Delta v}\right|=\sqrt{800}$

$\left|\overrightarrow{\Delta v}\right|=20\sqrt{2}m{s}^{-1}$

The truck is moving from south to north and then turns towards the west. Hence the direction is taken as southwest.

Thus,

Option D) $20\sqrt{2}m{s}^{-1}$ along southwest direction is the correct option.