Two particles are projected simultaneously from a point with different speeds and along the different directions. When both the particles are in air, then
what will be the path of one particle with respect to the other ?

Options

A. parabola
B. hyperbola
C. straight line
D. ellipse

Explanation

To solve the problem of determining the path of one particle with respect to the other when two particles are projected simultaneously from a point with different speeds and directions, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two particles, A and B, projected from the same point.
- Each particle has a different speed and is projected at different angles.

2. **Define the Velocities**:
- Let the velocity of particle A be \( \vec{u_a} \) and the velocity of particle B be \( \vec{u_b} \).
- The components of the velocities can be defined as:
- For particle A:
- Horizontal component: \( u_a \cos(\theta_a) \)
- Vertical component: \( u_a \sin(\theta_a) \)
- For particle B:
- Horizontal component: \( u_b \cos(\theta_b) \)
- Vertical component: \( u_b \sin(\theta_b) \)

3. **Define the Accelerations**:
- Both particles experience gravitational acceleration downwards, which can be represented as:
- \( \vec{g} = -g \hat{j} \)

4. **Relative Motion**:
- To find the path of one particle with respect to the other, we need to analyze their relative motion.
- The relative velocity of particle A with respect to particle B is given by:
\[ \vec{v_{AB}} = \vec{u_a} - \vec{u_b} \]
- The relative acceleration of particle A with respect to particle B is:
\[ \vec{a_{AB}} = \vec{a_a} - \vec{a_b} = 0 \]
- Since both particles have the same acceleration due to gravity, their relative acceleration is zero.

5. **Conclusion on Path**:
- Since the relative acceleration is zero, the relative velocity \( \vec{v_{AB}} \) remains constant.
- This means that particle A will appear to move in a straight line with respect to particle B, as there is no change in the relative velocity.

6. **Final Answer**:
- The path of one particle with respect to the other will be a straight line.