In the following questions a statement of assertion (A) is followed by a statement of reason ( R).
A: Path of a projectile with respect of another projectile is straight line .
R : Acceleration of a projectile with respect to another projectile is zero.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion then mark (1) .
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion then mark (2)
C. If Assertion is true statement but Reason is false then mark (3).
D. If both Assertion and Reason are false statements then mark (4) .

Answer

To solve the problem, we need to analyze the assertion (A) and the reason (R) provided in the question.

### Step 1: Understanding the Assertion
The assertion states that "the path of a projectile with respect to another projectile is a straight line."

### Step 2: Analyzing the Motion of Two Projectiles
Let’s consider two projectiles, P1 and P2.
- Let P1 have an initial speed \( u_1 \) and an angle of projection \( \theta_1 \).
- Let P2 have an initial speed \( u_2 \) and an angle of projection \( \theta_2 \).

The initial velocity of P1 can be broken down into its components:
- Horizontal component: \( u_{1x} = u_1 \cos(\theta_1) \)
- Vertical component: \( u_{1y} = u_1 \sin(\theta_1) \)

Similarly, for P2:
- Horizontal component: \( u_{2x} = u_2 \cos(\theta_2) \)
- Vertical component: \( u_{2y} = u_2 \sin(\theta_2) \)

### Step 3: Considering the Accelerations
Both projectiles are subject to the same gravitational acceleration \( g \) acting downwards. Therefore, the acceleration for both projectiles can be represented as:
- For P1: \( a_1 = -g \hat{j} \)
- For P2: \( a_2 = -g \hat{j} \)

### Step 4: Relative Motion of P1 with Respect to P2
To find the relative motion of P1 with respect to P2, we calculate the relative acceleration:
\[ a_{net} = a_1 - a_2 = (-g \hat{j}) - (-g \hat{j}) = 0 \]
This shows that the net acceleration of P1 with respect to P2 is zero.

### Step 5: Conclusion about the Path
Since the relative acceleration is zero, the relative velocity \( v_{relative} \) remains constant. This means that the path traced by P1 with respect to P2 will be a straight line.

### Step 6: Evaluating the Reason
The reason states that "the acceleration of a projectile with respect to another projectile is zero." This is indeed true as we have shown in Step 4.

### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion.

### Answer
Both assertion and reason are true, and the reason is the correct explanation of the assertion.

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