A truck travelling due to north at `20m s^(-1)` turns west and travels at the same speed. Find the change in its velocity.

Options

A. `40ms^(-1)N-W`
B. `20sqrt(2) ms^(-1) N-W`
C. `40 ms ^(-1)S-W`
D. `20sqrt(2) ms^(-1) S-W`

Explanation

To find the change in velocity of the truck, we will follow these steps:

### Step 1: Define the initial and final velocities
The truck is initially traveling due north at a speed of 20 m/s. We can represent this velocity as a vector:
- Initial velocity, \( \vec{v_1} = 20 \, \hat{j} \, \text{m/s} \) (where \( \hat{j} \) represents the north direction).

After turning west, the truck travels at the same speed of 20 m/s. We can represent this final velocity as:
- Final velocity, \( \vec{v_2} = -20 \, \hat{i} \, \text{m/s} \) (where \( \hat{i} \) represents the east direction, and the negative sign indicates west).

### Step 2: Calculate the change in velocity
The change in velocity \( \Delta \vec{v} \) can be calculated using the formula:
\[ \Delta \vec{v} = \vec{v_2} - \vec{v_1} \]
Substituting the values:
\[ \Delta \vec{v} = (-20 \, \hat{i}) - (20 \, \hat{j}) = -20 \, \hat{i} - 20 \, \hat{j} \]
This can be rewritten as:
\[ \Delta \vec{v} = -20 \, \hat{i} - 20 \, \hat{j} \]

### Step 3: Find the magnitude of the change in velocity
To find the magnitude of the change in velocity, we use the formula:
\[ |\Delta \vec{v}| = \sqrt{(-20)^2 + (-20)^2} \]
Calculating this gives:
\[ |\Delta \vec{v}| = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2} \, \text{m/s} \]

### Step 4: Determine the direction of the change in velocity
The direction of the change in velocity can be found using the components of the vector:
\[ \Delta \vec{v} = -20 \, \hat{i} - 20 \, \hat{j} \]
This indicates that the change in velocity is directed towards the southwest (since both components are negative).

To find the angle \( \theta \) with respect to the negative x-axis (west), we can use:
\[ \tan \theta = \frac{|\text{opposite}|}{|\text{adjacent}|} = \frac{20}{20} = 1 \]
Thus,
\[ \theta = \tan^{-1}(1) = 45^\circ \]
This angle indicates that the change in velocity is directed 45 degrees south of west.

### Final Answer
The change in velocity of the truck is:
\[ 20\sqrt{2} \, \text{m/s} \, \text{towards the southwest} \]