A car is moving on a straight road due north with a uniform speed of `50 km h^-1 `when it turns left through `90^@`. If the speed remains unchanged after turning, the change in the velocity of the car in the turning process is

Options

A. zero
B. `50sqrt(2) km h^(-1)` S-W direction
C. `50sqrt(2) km h^(-1)` N-W direction
D. `50sqrt(2) km h^(-1)` due to west

Answer

To solve the problem of finding the change in velocity of the car after it turns left through 90 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial and Final Velocities:**
- The car is initially moving due north with a speed of 50 km/h. We can represent this initial velocity as a vector:
\[ \vec{v_1} = 50 \, \text{km/h} \, \text{(north)} \]
- After turning left through 90 degrees, the car is now moving due west with the same speed. We represent this final velocity as:
\[ \vec{v_2} = 50 \, \text{km/h} \, \text{(west)} \]

2. **Calculate Change in Velocity:**
- The change in velocity (\(\Delta \vec{v}\)) is given by the difference between the final and initial velocities:
\[ \Delta \vec{v} = \vec{v_2} - \vec{v_1} \]

3. **Vector Representation:**
- To perform the subtraction, we need to represent these vectors in a coordinate system. We can use the following representation:
- North can be represented as (0, 50) in Cartesian coordinates.
- West can be represented as (-50, 0) in Cartesian coordinates.
- Thus, we have:
\[ \vec{v_1} = (0, 50) \quad \text{and} \quad \vec{v_2} = (-50, 0) \]

4. **Perform the Vector Subtraction:**
- Now we can calculate the change in velocity:
\[ \Delta \vec{v} = (-50, 0) - (0, 50) = (-50, -50) \]

5. **Magnitude of Change in Velocity:**
- To find the magnitude of the change in velocity, we use the Pythagorean theorem:
\[ |\Delta \vec{v}| = \sqrt{(-50)^2 + (-50)^2} = \sqrt{2500 + 2500} = \sqrt{5000} = 50\sqrt{2} \, \text{km/h} \]

6. **Direction of Change in Velocity:**
- The direction of the change in velocity vector \((-50, -50)\) is towards the southwest. This can be determined by recognizing that both components are negative, indicating a direction that is downwards and to the left from the origin.

### Final Result:
The change in velocity of the car after turning left through 90 degrees is:
\[ 50\sqrt{2} \, \text{km/h} \, \text{in the southwest direction.} \]