A car is moving along a road with a speed of `45km//hr`. In what direction must a body be projected form it with a velocity of `25m//s`, so that its resultant motion is at right angles to the direction of car?

Options

A. at an angle `120^(@)` with direction of motion of car.
B. at an angle `60^(@)` with direction of motion of car.
C. at an angle `90^(@)` with direction of motion of car.
D. at an angle `135^(@)` with direction of motion of car.

Explanation

To solve the problem, we need to determine the angle at which a body (stone) must be projected from a car moving at a speed of 45 km/h so that its resultant motion is perpendicular to the direction of the car.

### Step-by-Step Solution:

1. **Convert the speed of the car from km/h to m/s**:
\[ \text{Speed of car} = 45 \text{ km/h} = \frac{45 \times 1000}{3600} = 12.5 \text{ m/s} \]

2. **Identify the speed of the stone**:
The speed of the stone is given as 25 m/s.

3. **Set up the vector components**:
- Let the car's velocity vector be \( \vec{V_c} = 12.5 \hat{i} \) (along the x-axis).
- Let the stone's velocity vector be \( \vec{V_s} \) which can be expressed in terms of its magnitude and angle \( \theta \):
\[ \vec{V_s} = 25 \cos(\theta) \hat{i} + 25 \sin(\theta) \hat{j} \]

4. **Find the resultant velocity**:
The resultant velocity \( \vec{V_r} \) is given by:
\[ \vec{V_r} = \vec{V_c} + \vec{V_s} = (12.5 + 25 \cos(\theta)) \hat{i} + (25 \sin(\theta)) \hat{j} \]

5. **Condition for perpendicularity**:
For the resultant velocity to be perpendicular to the car's velocity, the dot product of \( \vec{V_r} \) with \( \vec{V_c} \) must be zero:
\[ (12.5 + 25 \cos(\theta)) \cdot 12.5 + (25 \sin(\theta)) \cdot 0 = 0 \]
Simplifying this gives:
\[ 12.5 + 25 \cos(\theta) = 0 \]

6. **Solve for \( \cos(\theta) \)**:
\[ 25 \cos(\theta) = -12.5 \implies \cos(\theta) = -\frac{12.5}{25} = -\frac{1}{2} \]

7. **Find the angle \( \theta \)**:
The angle \( \theta \) for which \( \cos(\theta) = -\frac{1}{2} \) is:
\[ \theta = 120^\circ \quad \text{(in the second quadrant)} \]

### Final Answer:
The body must be projected at an angle of \( 120^\circ \) from the direction of the car's motion.