JEE PYQ: Motion in a Plane - Question ID ed1c3121c35a (JEE Main 2022)

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 initial velocity $v_0$ at an angle $\theta$ with the horizontal is:

$R = \frac{v_0^2 \sin(2\theta)}{g}$

where:

• $v_0$ is the initial velocity (same for both projectiles in this question),
• $\theta$ is the launch angle,
• $g$ is the acceleration due to gravity.

This formula arises from combining the horizontal and vertical components of motion and solving for the time of flight and horizontal displacement.

Step-by-Step Derivation:

We are given two projectiles with the same initial velocity $v_0$, launched at angles $\theta_1 = 45^\circ$ and $\theta_2 = 30^\circ$. We need to find the ratio of their ranges, $R_1 : R_2$.

Step 1: Write the range formula for both projectiles.

For the first projectile (angle $45^\circ$): $R_1 = \frac{v_0^2 \sin(2 \cdot 45^\circ)}{g} = \frac{v_0^2 \sin(90^\circ)}{g} = \frac{v_0^2 \cdot 1}{g} = \frac{v_0^2}{g}$ For the second projectile (angle $30^\circ$): $R_2 = \frac{v_0^2 \sin(2 \cdot 30^\circ)}{g} = \frac{v_0^2 \sin(60^\circ)}{g} = \frac{v_0^2 \cdot \frac{\sqrt{3}}{2}}{g} = \frac{v_0^2 \sqrt{3}}{2g}$

Step 2: Compute the ratio $R_1 : R_2$.

$\frac{R_1}{R_2} = \frac{\frac{v_0^2}{g}}{\frac{v_0^2 \sqrt{3}}{2g}} = \frac{v_0^2}{g} \cdot \frac{2g}{v_0^2 \sqrt{3}} = \frac{2}{\sqrt{3}}$

Step 3: Rationalize and express the ratio.

To express this ratio in a standard form, we rationalize the denominator: $\frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$ However, the question asks for the ratio $R_1 : R_2$, which we can write as: $R_1 : R_2 = 2 : \sqrt{3}$

Step 4: Match with the given options.

The derived ratio $2 : \sqrt{3}$ matches option C. Common Traps & Exam Tip:

Trap 1: Misapplying the range formula. Some students mistakenly use $\sin \theta$ or $\cos \theta$ instead of $\sin(2\theta)$ in the range formula. Always remember that the range depends on $\sin(2\theta)$, not just $\sin \theta$ or $\cos \theta$.

Trap 2: Forgetting to double the angle. A common error is to compute $\sin \theta$ instead of $\sin(2\theta)$. For example, calculating $\sin(30^\circ) = 0.5$ instead of $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ leads to an incorrect ratio.

Trap 3: Incorrect simplification. When simplifying the ratio, students may forget to cancel out $v_0^2$ and $g$, leading to unnecessary complexity. Always cancel common terms early to simplify calculations.

Exam Tip: For problems involving ratios of trigonometric functions, it is often helpful to evaluate the trigonometric values first (e.g., $\sin(90^\circ) = 1$, $\sin(60^\circ) = \frac{\sqrt{3}}{2}$) before substituting into the formula. This reduces the chance of algebraic errors.