From a tower of height `40 m`, two bodies are simultaneously projected horizontally in opposite direction, with velocities `2 m s^-1 and 8 ms^-1`. respectively.
The time taken for the velocity vectors of two bodies to become perpendicular to each other is :

Options

A. 0.1 s
B. 0.2 s
C. 0.4 s
D. 0.8 s

Explanation

To solve the problem, we need to determine the time at which the velocity vectors of the two bodies become perpendicular to each other after being projected horizontally from a height of 40 m.

### Step-by-step Solution:

1. **Identify the initial conditions**:
- Body 1 is projected horizontally with a velocity \( v_{1x} = 2 \, \text{m/s} \) to the right.
- Body 2 is projected horizontally with a velocity \( v_{2x} = 8 \, \text{m/s} \) to the left.
- Both bodies are projected from a height of \( h = 40 \, \text{m} \).

2. **Determine the vertical velocity components**:
- For both bodies, the initial vertical velocity \( v_{y} \) is \( 0 \, \text{m/s} \).
- The vertical velocity for both bodies after time \( t \) is given by the equation:
\[ v_{y} = u_{y} + a_{y} \cdot t = 0 - g \cdot t = -g \cdot t \]
- Here, \( g \) is the acceleration due to gravity, approximately \( 10 \, \text{m/s}^2 \).

3. **Write the velocity vectors**:
- The velocity vector of Body 1:
\[ \mathbf{V_1} = 2 \hat{i} - g t \hat{j} \]
- The velocity vector of Body 2:
\[ \mathbf{V_2} = -8 \hat{i} - g t \hat{j} \]

4. **Condition for perpendicularity**:
- Two vectors are perpendicular if their dot product is zero:
\[ \mathbf{V_1} \cdot \mathbf{V_2} = 0 \]
- Calculating the dot product:
\[ (2 \hat{i} - g t \hat{j}) \cdot (-8 \hat{i} - g t \hat{j}) = 2 \cdot (-8) + (-g t) \cdot (-g t) \]
\[ = -16 + g^2 t^2 = 0 \]

5. **Solve for time \( t \)**:
- Rearranging the equation:
\[ g^2 t^2 = 16 \]
- Taking the square root:
\[ t^2 = \frac{16}{g^2} \]
- Therefore:
\[ t = \frac{4}{g} \]
- Substituting \( g \approx 10 \, \text{m/s}^2 \):
\[ t = \frac{4}{10} = 0.4 \, \text{s} \]

### Final Answer:
The time taken for the velocity vectors of the two bodies to become perpendicular to each other is \( 0.4 \, \text{s} \).

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