Two particles are projected simultaneously in the same vertical plane from the same point, with different speeds `u_(1)` and `u_(2)`, making angles `theta_(1)` and `theta_(2)` respectively with the horizontal, such that `u_(1) cos theta_(1) = u_(2) cos theta_(2)`. The path followed by one, as seen by the 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 `(ne 90^(@))` with horizontal

Explanation

To solve the problem, we need to analyze the motion of two particles projected simultaneously from the same point with different speeds and angles. The key condition given is that the horizontal components of their velocities are equal.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Two particles are projected with speeds \( u_1 \) and \( u_2 \).
- They make angles \( \theta_1 \) and \( \theta_2 \) with the horizontal.
- The condition given is \( u_1 \cos \theta_1 = u_2 \cos \theta_2 \).

2. **Determine the Velocity Components**:
- For Particle 1:
- Horizontal velocity: \( v_{1x} = u_1 \cos \theta_1 \)
- Vertical velocity: \( v_{1y} = u_1 \sin \theta_1 - gt \)
- For Particle 2:
- Horizontal velocity: \( v_{2x} = u_2 \cos \theta_2 \)
- Vertical velocity: \( v_{2y} = u_2 \sin \theta_2 - gt \)

3. **Use the Given Condition**:
- Since \( u_1 \cos \theta_1 = u_2 \cos \theta_2 \), we can denote this common horizontal velocity as \( v_x \). Thus, both particles have the same horizontal velocity.

4. **Calculate the Relative Velocity**:
- The relative velocity of Particle 1 with respect to Particle 2 is given by:
\[ v_{rel} = v_1 - v_2 \]
- This results in:
\[ v_{rel,x} = v_{1x} - v_{2x} = 0 \quad \text{(since they have the same horizontal component)} \]
- For the vertical component:
\[ v_{rel,y} = v_{1y} - v_{2y} = (u_1 \sin \theta_1 - gt) - (u_2 \sin \theta_2 - gt) = u_1 \sin \theta_1 - u_2 \sin \theta_2 \]

5. **Analyze the Path**:
- The relative vertical velocity \( v_{rel,y} = u_1 \sin \theta_1 - u_2 \sin \theta_2 \) is a constant (since \( u_1 \) and \( u_2 \) are constants).
- Since the horizontal component of the relative velocity is zero, the motion of one particle as seen from the other will be purely vertical.

6. **Conclusion**:
- The path followed by one particle as seen by the other is a straight line making a constant angle with the horizontal (not equal to 90 degrees).

### Final Answer:
The path followed by one particle as seen by the other is a straight line making a constant angle not equal to 90 degrees with the horizontal.

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