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
The trajectory of particle `1` with respect to particle `2` wil be

Options

A. a parabola
B. a straight line
C. a vertical striaght line
D. a horizontal straight line

Explanation

To determine the trajectory of particle 1 with respect to particle 2, we can follow these steps:

### Step 1: Understand the Motion of Each Projectile
Both projectiles are thrown simultaneously from the same point but with different velocities and angles. Projectile 1 has a velocity \( v_1 \) at an angle \( \theta_1 \) and projectile 2 has a velocity \( v_2 \) at an angle \( \theta_2 \).

### Step 2: Analyze the Frame of Reference
When we analyze the motion of one projectile with respect to the other, we need to consider the frame of reference. If we are observing the motion from the perspective of projectile 1, we are in a non-inertial frame of reference because projectile 1 is accelerating due to gravity.

### Step 3: Determine the Acceleration of Each Projectile
Both projectiles experience the same gravitational acceleration downward, which is \( g \). Therefore, the net force acting on each projectile is equal to their respective weights, and both will have an acceleration of \( g \) downward.

### Step 4: Relative Motion
In the frame of reference of projectile 1, projectile 2 will appear to move relative to it. Since both projectiles are under the influence of gravity, the vertical component of their velocities will change over time, but the horizontal component will remain constant.

### Step 5: Trajectory of Particle 1 with Respect to Particle 2
Since both projectiles are accelerating downwards at the same rate, the relative motion will be such that projectile 2 will appear to move in a straight line with respect to projectile 1. This is because the acceleration of both projectiles is the same, which means that the trajectory of particle 1 with respect to particle 2 will be a straight line.

### Conclusion
The trajectory of particle 1 with respect to particle 2 will be a straight line.

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