Two particles are projected simultaneously in the same vertical plane, with speed `u_(1)` and `u_(2)` at angle of projection `theta_(1)` and `theta_(2)` respectively with the horizontal. The path followed by one, as seen by other (as long as both are in flight), is

Options

A. a vertical line
B. a parabola
C. a hyperbola
D. a straight line making a constant angle with the horizontal

Answer

To solve the problem of the paths followed by two particles projected simultaneously in the same vertical plane, we can break down the solution into clear steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Let the first particle have an initial speed \( u_1 \) and be projected at an angle \( \theta_1 \) with the horizontal.
- Let the second particle have an initial speed \( u_2 \) and be projected at an angle \( \theta_2 \) with the horizontal.

2. **Resolve the Velocities into Components**:
- For the first particle:
- Horizontal component: \( u_{1x} = u_1 \cos \theta_1 \)
- Vertical component: \( u_{1y} = u_1 \sin \theta_1 \)
- For the second particle:
- Horizontal component: \( u_{2x} = u_2 \cos \theta_2 \)
- Vertical component: \( u_{2y} = u_2 \sin \theta_2 \)

3. **Determine the Velocities After Time \( t \)**:
- The horizontal velocity remains constant for both particles since there is no horizontal acceleration:
- \( v_{1x} = u_{1x} = u_1 \cos \theta_1 \)
- \( v_{2x} = u_{2x} = u_2 \cos \theta_2 \)
- The vertical velocity changes due to gravitational acceleration \( g \):
- \( v_{1y} = u_{1y} - gt = u_1 \sin \theta_1 - gt \)
- \( v_{2y} = u_{2y} - gt = u_2 \sin \theta_2 - gt \)

4. **Express the Velocities in Vector Form**:
- For the first particle:
\[ \mathbf{V_1} = (u_1 \cos \theta_1) \hat{i} + (u_1 \sin \theta_1 - gt) \hat{j} \]
- For the second particle:
\[ \mathbf{V_2} = (u_2 \cos \theta_2) \hat{i} + (u_2 \sin \theta_2 - gt) \hat{j} \]

5. **Find the Relative Velocity**:
- The relative velocity of particle 1 as seen from particle 2 is given by:
\[ \mathbf{V_{12}} = \mathbf{V_1} - \mathbf{V_2} \]
- Substituting the expressions for \( \mathbf{V_1} \) and \( \mathbf{V_2} \):
\[ \mathbf{V_{12}} = \left( u_1 \cos \theta_1 - u_2 \cos \theta_2 \right) \hat{i} + \left( u_1 \sin \theta_1 - u_2 \sin \theta_2 \right) \hat{j} \]

6. **Analyze the Result**:
- The relative velocity \( \mathbf{V_{12}} \) is a constant vector because it does not depend on time \( t \).
- Since the relative velocity is constant, the path followed by one particle as seen by the other will be a straight line.

7. **Conclusion**:
- Therefore, the path followed by one particle as seen by the other is a straight line making a constant angle with the horizontal.

### Final Answer:
The path followed by one particle as seen by the other is a straight line.