A particle is moving with velocity 5 m/s towards cast and its velocity changes to 5 m/s north in 10 s, find the acceleration .

Options

A. `sqrt(2) ` North- West
B. `(1)/(sqrt(2))` North - West
C. `(1)/(sqrt(2))` North - East
D. `sqrt(2)` North -East

Explanation

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

### Step 1: Identify Initial and Final Velocities
The particle is initially moving towards the east with a velocity of:
- **Initial Velocity (u)** = 5 m/s (east) = 5 î m/s

After 10 seconds, it moves north with a velocity of:
- **Final Velocity (v)** = 5 m/s (north) = 5 ĵ m/s

### Step 2: Calculate Change in Velocity
The change in velocity (Δv) can be calculated as:
\[ \Delta v = v - u = (5 ĵ - 5 î) \text{ m/s} = -5 î + 5 ĵ \text{ m/s} \]

### Step 3: Calculate Acceleration
Acceleration (a) is defined as the change in velocity over time:
\[ a = \frac{\Delta v}{t} \]
Substituting the values:
\[ a = \frac{-5 î + 5 ĵ}{10 \text{ s}} = -0.5 î + 0.5 ĵ \text{ m/s}^2 \]

### Step 4: Find the Magnitude of Acceleration
To find the magnitude of the acceleration vector:
\[ |a| = \sqrt{(-0.5)^2 + (0.5)^2} = \sqrt{0.25 + 0.25} = \sqrt{0.5} = \frac{1}{\sqrt{2}} \text{ m/s}^2 \]

### Step 5: Determine the Direction of Acceleration
The direction of the acceleration vector can be determined from its components:
- The x-component is negative (-0.5), indicating a westward direction.
- The y-component is positive (0.5), indicating a northward direction.

Thus, the acceleration is directed towards the northwest.

### Final Answer
The acceleration of the particle is:
\[ \frac{1}{\sqrt{2}} \text{ m/s}^2 \text{ towards the northwest.} \]

---