A particle is moving eastwards with a velocity of ` 5 ms_(-1)`. In `10 seconds` the velocity changes to `5 ms^(-1)` northwards. The average acceleration in this time is

Options

A. `(1)/(2) ms^(-2)` towards north
B. `(1)/ (sqrt(2 ms^(-2)))` towards north - east
C. `(1)/ (sqrt (2 ms^(-2)))` towards north -west
D. ` zero`

Explanation

To find the average acceleration of the particle, we can follow these steps:

### Step 1: Understand the Initial and Final Velocities
The particle is initially moving eastwards with a velocity of \( \mathbf{v_i} = 5 \, \text{m/s} \) in the positive x-direction. This can be represented as:
\[ \mathbf{v_i} = 5 \, \hat{i} \, \text{m/s} \]
After 10 seconds, the particle is moving northwards with a velocity of \( \mathbf{v_f} = 5 \, \text{m/s} \) in the positive y-direction. This can be represented as:
\[ \mathbf{v_f} = 5 \, \hat{j} \, \text{m/s} \]

### Step 2: Calculate the Change in Velocity
The change in velocity \( \Delta \mathbf{v} \) can be calculated as:
\[ \Delta \mathbf{v} = \mathbf{v_f} - \mathbf{v_i} \]
Substituting the values:
\[ \Delta \mathbf{v} = (5 \, \hat{j}) - (5 \, \hat{i}) = -5 \, \hat{i} + 5 \, \hat{j} \]

### Step 3: Calculate the Magnitude of the Change in Velocity
The magnitude of \( \Delta \mathbf{v} \) can be calculated using the Pythagorean theorem:
\[ |\Delta \mathbf{v}| = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \, \text{m/s} \]

### Step 4: Calculate the Average Acceleration
The average acceleration \( \mathbf{a_{avg}} \) is given by the formula:
\[ \mathbf{a_{avg}} = \frac{\Delta \mathbf{v}}{\Delta t} \]
Where \( \Delta t = 10 \, \text{s} \). Substituting the values:
\[ \mathbf{a_{avg}} = \frac{5\sqrt{2} \, \text{m/s}}{10 \, \text{s}} = \frac{\sqrt{2}}{2} \, \text{m/s}^2 \]

### Step 5: Final Result
Thus, the magnitude of the average acceleration is:
\[ |\mathbf{a_{avg}}| = \frac{\sqrt{2}}{2} \, \text{m/s}^2 \]

### Summary
The average acceleration of the particle over the 10 seconds is \( \frac{\sqrt{2}}{2} \, \text{m/s}^2 \).

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