Two projectiles are thrown simultaneously in the same plane from the same point. If their velocities are `v_(1)` and `v_(2)` at angles `theta_(1)` and `theta_(2)` respectively from the horizontal, then ansewer the following questions
If `v_(1)costheta_(1) = v_(2)cos theta_(2)`, then choose the incrorrect statement

Options

A. one particle will remain exactly below or above the other particle
B. the trajectory of one with respect to other will be a vertical straight line
C. both will have the same range
D. none of these

Explanation

To solve the problem, we need to analyze the given conditions and the statements provided about the two projectiles.

### Given:
- Two projectiles are thrown simultaneously from the same point.
- Their velocities are \( v_1 \) and \( v_2 \) at angles \( \theta_1 \) and \( \theta_2 \) respectively.
- It is given that \( v_1 \cos \theta_1 = v_2 \cos \theta_2 \).

### Objective:
Identify the incorrect statement based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Horizontal Components:**
The condition \( v_1 \cos \theta_1 = v_2 \cos \theta_2 \) indicates that the horizontal components of the velocities of both projectiles are equal. This means both projectiles will have the same horizontal displacement over time.

**Hint:** Remember that the horizontal component of velocity affects how far the projectile travels horizontally.

2. **Analyzing the Vertical Motion:**
The vertical components of the velocities are given by \( v_{1y} = v_1 \sin \theta_1 \) and \( v_{2y} = v_2 \sin \theta_2 \). The time of flight for each projectile can be calculated using the vertical motion equations, which depend on the vertical components of their velocities.

**Hint:** The time of flight is influenced by the vertical component of the velocity and the acceleration due to gravity.

3. **Time of Flight Comparison:**
Since the horizontal components are equal, we can conclude that if the time of flight is the same, the horizontal distance covered will also be the same. However, the time of flight may not necessarily be the same unless the vertical components are also equal.

**Hint:** Check if the vertical components lead to the same time of flight.

4. **Maximum Height:**
The maximum height attained by each projectile can be calculated using the vertical components. If the vertical components are equal, then the maximum heights will also be equal.

**Hint:** Maximum height depends on the vertical component of the initial velocity.

5. **Trajectory with Respect to Each Other:**
Since the horizontal components are equal, the trajectory of one projectile with respect to the other will appear as a vertical straight line when viewed from the frame of reference of one of the projectiles.

**Hint:** Think about how the relative motion looks when two objects have the same horizontal speed.

6. **Identifying the Incorrect Statement:**
The statements can be analyzed:
- If the horizontal components are equal, the trajectories will be vertical lines.
- The time of flight is not guaranteed to be the same unless the vertical components are also equal.
- The maximum heights can be equal if the vertical components are equal.

The incorrect statement is likely to be one that assumes equal ranges or equal times of flight without confirming the vertical components.

### Conclusion:
The incorrect statement is that both projectiles will have the same range. This is because the ranges depend on both the horizontal and vertical components of the velocities, and while the horizontal components are equal, the vertical components may not be.

### Final Answer:
The incorrect statement is: "Both projectiles will have the same range."