A car is moving eastward with velocity 10 m/s. In 20 sec, the velocity change to 10 m/s northwards. The averge acceleration in this time.

Options

A. `1//sqrt(2) m//s^(2)` toward N-W
B. `1//sqrt(2) m//s^(2)` towards N-E
C. `1//sqrt(2) m//s^(2)` towards N-W
D. `1//sqrt(2) m//s^(2)` towards N

Explanation

To solve the problem of finding the average acceleration of a car that changes its velocity from 10 m/s eastward to 10 m/s northward over a time period of 20 seconds, we can follow these steps:

### Step 1: Identify the Initial and Final Velocities
- The initial velocity (\( \vec{v_i} \)) is 10 m/s eastward, which can be represented as:
\[ \vec{v_i} = 10 \hat{i} \, \text{m/s} \]
- The final velocity (\( \vec{v_f} \)) is 10 m/s northward, represented as:
\[ \vec{v_f} = 10 \hat{j} \, \text{m/s} \]

### Step 2: Calculate the Change in Velocity
- The change in velocity (\( \Delta \vec{v} \)) is given by:
\[ \Delta \vec{v} = \vec{v_f} - \vec{v_i} \]
- Substituting the values:
\[ \Delta \vec{v} = (10 \hat{j} - 10 \hat{i}) \, \text{m/s} = -10 \hat{i} + 10 \hat{j} \, \text{m/s} \]

### Step 3: Calculate the Magnitude of the Change in Velocity
- The magnitude of the change in velocity can be calculated using the Pythagorean theorem:
\[ |\Delta \vec{v}| = \sqrt{(-10)^2 + (10)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{m/s} \]

### Step 4: Calculate the Average Acceleration
- The average acceleration (\( \vec{a_{avg}} \)) is defined as the change in velocity divided by the change in time:
\[ \vec{a_{avg}} = \frac{\Delta \vec{v}}{\Delta t} \]
- Given that the change in time (\( \Delta t \)) is 20 seconds:
\[ \vec{a_{avg}} = \frac{-10 \hat{i} + 10 \hat{j}}{20} = -0.5 \hat{i} + 0.5 \hat{j} \, \text{m/s}^2 \]

### Step 5: Calculate the Magnitude of Average Acceleration
- The magnitude of the average acceleration can be calculated as:
\[ |\vec{a_{avg}}| = \sqrt{(-0.5)^2 + (0.5)^2} = \sqrt{0.25 + 0.25} = \sqrt{0.5} = \frac{1}{\sqrt{2}} \, \text{m/s}^2 \]

### Step 6: Determine the Direction of Average Acceleration
- The direction of the average acceleration is in the northwest direction since it has components in both the negative x-direction (west) and positive y-direction (north).

### Final Result
The average acceleration of the car is:
\[ |\vec{a_{avg}}| = \frac{1}{\sqrt{2}} \, \text{m/s}^2 \quad \text{(in the northwest direction)} \]

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