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 linc
C. a vertical straight line
D. a horizontal straight line

Answer

To solve the problem of determining the trajectory of particle 1 with respect to particle 2, we will analyze the motion of both projectiles and their relative motion. Here’s a step-by-step solution:

### Step 1: Resolve the velocities of both projectiles
For projectile 1, with initial velocity \( v_1 \) at angle \( \theta_1 \):
- Horizontal component of velocity: \( v_{1x} = v_1 \cos \theta_1 \)
- Vertical component of velocity: \( v_{1y} = v_1 \sin \theta_1 \)

For projectile 2, with initial velocity \( v_2 \) at angle \( \theta_2 \):
- Horizontal component of velocity: \( v_{2x} = v_2 \cos \theta_2 \)
- Vertical component of velocity: \( v_{2y} = v_2 \sin \theta_2 \)

### Step 2: Write the equations of motion for both projectiles
The position of projectile 1 as a function of time \( t \) can be expressed as:
- \( x_1(t) = v_{1x} t = v_1 \cos \theta_1 \cdot t \)
- \( y_1(t) = v_{1y} t - \frac{1}{2} g t^2 = v_1 \sin \theta_1 \cdot t - \frac{1}{2} g t^2 \)

The position of projectile 2 can be expressed as:
- \( x_2(t) = v_{2x} t = v_2 \cos \theta_2 \cdot t \)
- \( y_2(t) = v_{2y} t - \frac{1}{2} g t^2 = v_2 \sin \theta_2 \cdot t - \frac{1}{2} g t^2 \)

### Step 3: Determine the relative position of particle 1 with respect to particle 2
The relative position can be defined as:
- \( x_{rel}(t) = x_1(t) - x_2(t) = (v_1 \cos \theta_1 - v_2 \cos \theta_2) t \)
- \( y_{rel}(t) = y_1(t) - y_2(t) = (v_1 \sin \theta_1 - v_2 \sin \theta_2) t \)

### Step 4: Analyze the relative motion
The relative motion can be expressed as:
- \( x_{rel}(t) = (v_1 \cos \theta_1 - v_2 \cos \theta_2) t \)
- \( y_{rel}(t) = (v_1 \sin \theta_1 - v_2 \sin \theta_2) t - \frac{1}{2} g t^2 \)

### Step 5: Determine the nature of the trajectory
Since the acceleration due to gravity affects both projectiles equally (both have the same vertical acceleration of \( -g \)), the relative acceleration between the two projectiles is zero. This means that the relative velocity is constant over time.

### Conclusion
Since the relative velocity is constant and the motion is linear, the trajectory of particle 1 with respect to particle 2 will be a straight line.

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

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