JEE PYQ: Motion in a Plane - Question ID 36d52c23b872 (JEE Main 2023)

Core Formula & Concept:

In projectile motion, the range $R$ is the horizontal distance traveled by the projectile before it returns to the same vertical level from which it was launched. The key formula for the range of a projectile launched with speed $u$ at an angle $\theta$ to the horizontal (ignoring air resistance) is:

$R = \frac{u^2 \sin 2\theta}{g}.$

Here:

• $u$ is the initial speed,
• $\theta$ is the projection angle,
• $g$ is the acceleration due to gravity.

The factor $\sin 2\theta$ determines how the range varies with the angle $\theta$. Since $\sin 2\theta$ has a maximum value of $1$, the range $R$ is maximized when $\sin 2\theta = 1$.

Step-by-Step Derivation:

Step 1: Recall the range formula
The horizontal range of a projectile is $R = \frac{u^2 \sin 2\theta}{g}.$

Step 2: Maximize the range
To find the angle $\theta$ that gives the maximum range, we maximize $\sin 2\theta$. The sine function reaches its maximum value of $1$ when its argument is $90^\circ$. Therefore, $\sin 2\theta = 1 \quad\Longrightarrow\quad 2\theta = 90^\circ \quad\Longrightarrow\quad \theta = 45^\circ.$

Step 3: Interpret Assertion A and Reason R

• Assertion A states: “When a body is projected at an angle $45^\circ$, its range is maximum.” This is exactly what we derived in Step 2.
• Reason R states: “For maximum range, the value of $\sin 2\theta$ should be equal to one.” This is the mathematical condition we used to arrive at $\theta = 45^\circ$.

Step 4: Match with the given options
Both Assertion A and Reason R are true, and Reason R correctly explains why Assertion A is true. Hence the correct choice is A: Both A and R are correct and R is the correct explanation of A.

Common Traps & Exam Tip:

1. Confusing $\sin \theta$ with $\sin 2\theta$: Some students mistakenly try to maximize $\sin \theta$ instead of $\sin 2\theta$, leading them to an incorrect angle of $90^\circ$.
2. Forgetting the double-angle identity: Without recalling that $\sin 2\theta = 2\sin\theta\cos\theta$, one might not see why $45^\circ$ is special.
3. Overlooking the “explanation” part: Even if both statements are true, one must check whether R actually explains A. In this case it does, so option A is correct.