A truck travelling due north at 20 m/s turns east and travels at the same speed. What is the change in velocity :

Options

A. 40 m/s north east
B. `20 sqrt(2)` m/s south east
C. `20sqrt(2)` m/s south west
D. `20sqrt(2)` m/s north west

Answer

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

### Step 1: Identify the Initial and Final Velocities
- The truck is initially traveling due north at a speed of 20 m/s. We can represent this initial velocity vector as:
\[ \vec{u} = 20 \, \text{m/s} \, \text{(North)} \]
- After turning, the truck travels due east at the same speed of 20 m/s. We can represent this final velocity vector as:
\[ \vec{v} = 20 \, \text{m/s} \, \text{(East)} \]

### Step 2: Represent the Velocities as Vectors
- We can represent the north direction as the positive y-axis and the east direction as the positive x-axis. Thus:
- \(\vec{u} = 0 \hat{i} + 20 \hat{j}\) (where \(\hat{i}\) is the unit vector in the east direction and \(\hat{j}\) is the unit vector in the north direction)
- \(\vec{v} = 20 \hat{i} + 0 \hat{j}\)

### Step 3: Calculate the Change in Velocity
- The change in velocity (\(\Delta \vec{v}\)) can be calculated using the formula:
\[ \Delta \vec{v} = \vec{v} - \vec{u} \]
- Substituting the vectors:
\[ \Delta \vec{v} = (20 \hat{i} + 0 \hat{j}) - (0 \hat{i} + 20 \hat{j}) = 20 \hat{i} - 20 \hat{j} \]
- This simplifies to:
\[ \Delta \vec{v} = 20 \hat{i} - 20 \hat{j} \]

### Step 4: Calculate the Magnitude of the Change in Velocity
- To find the magnitude of the change in velocity, we can use the Pythagorean theorem:
\[ |\Delta \vec{v}| = \sqrt{(20)^2 + (-20)^2} = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2} \, \text{m/s} \]

### Step 5: Determine the Direction of the Change in Velocity
- The direction of the change in velocity can be found using the angle \(\theta\) with respect to the east direction (x-axis):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-20}{20} = -1 \]
- This gives us \(\theta = -45^\circ\), indicating that the change in velocity is directed towards the southeast.

### Final Answer
- The change in velocity of the truck is:
\[ |\Delta \vec{v}| = 20\sqrt{2} \, \text{m/s} \, \text{(southeast)} \]

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