JEE PYQ: Motion in a Plane - Question ID 2f3bf5ab1230 (JEE Main 2023)

Core Formula & Concept:

In projectile motion, the maximum height ($H$) attained by a projectile depends on its initial velocity ($u$), the angle of projection ($\theta$) with the horizontal, and the acceleration due to gravity ($g$). The key formula for maximum height is derived from the vertical component of the motion.

The vertical component of the initial velocity is $u_y = u \sin \theta$. At the maximum height, the vertical component of velocity becomes zero. Using the kinematic equation: $v_y^2 = u_y^2 - 2gH$ At maximum height, $v_y = 0$, so: $0 = (u \sin \theta)^2 - 2gH$ Solving for $H$: $H = \frac{u^2 \sin^2 \theta}{2g}$ This formula is central to solving the problem.

Step-by-Step Derivation:

We are given two projectiles projected at angles $30^\circ$ and $60^\circ$ with the same initial speed $u$. We need to find the ratio of their maximum heights.

Step 1: Write the maximum height formula for both projectiles.
For the first projectile (angle $30^\circ$): $H_1 = \frac{u^2 \sin^2 30^\circ}{2g}$ For the second projectile (angle $60^\circ$): $H_2 = \frac{u^2 \sin^2 60^\circ}{2g}$

Step 2: Compute $\sin 30^\circ$ and $\sin 60^\circ$.
We know: $\sin 30^\circ = \frac{1}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}$ Substitute these values into the height formulas: $H_1 = \frac{u^2 \left(\frac{1}{2}\right)^2}{2g} = \frac{u^2 \cdot \frac{1}{4}}{2g} = \frac{u^2}{8g}$ $H_2 = \frac{u^2 \left(\frac{\sqrt{3}}{2}\right)^2}{2g} = \frac{u^2 \cdot \frac{3}{4}}{2g} = \frac{3u^2}{8g}$

Step 3: Find the ratio $H_1 : H_2$.
Divide $H_1$ by $H_2$: $\frac{H_1}{H_2} = \frac{\frac{u^2}{8g}}{\frac{3u^2}{8g}} = \frac{1}{3}$ Thus, the ratio of the maximum heights is $1 : 3$.

Step 4: Match with the given options.
The correct ratio is $1 : 3$, which corresponds to option C.

Common Traps & Exam Tip:

1. Confusing $\sin \theta$ with $\cos \theta$: Some students mistakenly use the horizontal component ($u \cos \theta$) instead of the vertical component ($u \sin \theta$) when calculating maximum height. Always remember that maximum height depends on the vertical motion.

2. Incorrect squaring of trigonometric functions: A common error is to write $\sin^2 \theta$ as $\sin \theta^2$, which is incorrect. Ensure the square is applied to the entire sine function, not just the angle.

3. Forgetting to cancel common terms: In the ratio calculation, students sometimes forget to cancel $u^2$ and $g$, leading to unnecessary complexity. Always simplify expressions before computing ratios.

4. Misinterpreting the ratio order: The question asks for the ratio of the heights of the $30^\circ$ projectile to the $60^\circ$ projectile. Some students reverse the order, leading to the incorrect option $3 : 1$. Pay close attention to the order specified in the question.

Exam Tip: For problems involving ratios of trigonometric functions, it is often helpful to compute the numerical values of the trigonometric terms first (e.g., $\sin 30^\circ = 0.5$) before substituting into formulas. This reduces the chance of algebraic errors.