JEE PYQ: Motion in a Plane - Question ID d0148f71eabb (JEE Main 2026)

Core Formula & Concept:
This problem deals with the fundamental concept of projectile motion. When a projectile is launched with an initial velocity $u$ at an angle $\theta$ with respect to the horizontal, its trajectory is parabolic under the influence of gravity, assuming air resistance is negligible. One of the key parameters describing this motion is the horizontal range, which is the total horizontal distance covered by the projectile before it returns to the same horizontal level from which it was launched. The formula for the horizontal range ($R$) of a projectile is given by: $R = \frac{u^2 \sin(2\theta)}{g}$ where:

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

In this problem, the initial velocities ($u$) for both projectiles are the same, and $g$ is a constant. Therefore, the ratio of the ranges will depend solely on the $\sin(2\theta)$ term.
Step-by-Step Derivation:
Let the initial velocity for both projectiles be $u$. The acceleration due to gravity is $g$. For the first projectile: The angle of projection is $\theta_1 = 15^{\circ}$. Using the range formula, its range $R_1$ will be: $R_1 = \frac{u^2 \sin(2\theta_1)}{g}$ $R_1 = \frac{u^2 \sin(2 \times 15^{\circ})}{g}$ $R_1 = \frac{u^2 \sin(30^{\circ})}{g}$ For the second projectile: The angle of projection is $\theta_2 = 30^{\circ}$. Using the range formula, its range $R_2$ will be: $R_2 = \frac{u^2 \sin(2\theta_2)}{g}$ $R_2 = \frac{u^2 \sin(2 \times 30^{\circ})}{g}$ $R_2 = \frac{u^2 \sin(60^{\circ})}{g}$ Now, we need to find the ratio of their ranges, $R_1 : R_2$. $\frac{R_1}{R_2} = \frac{\frac{u^2 \sin(30^{\circ})}{g}}{\frac{u^2 \sin(60^{\circ})}{g}}$ Notice that the term $\frac{u^2}{g}$ is common in both the numerator and the denominator, so it cancels out. $\frac{R_1}{R_2} = \frac{\sin(30^{\circ})}{\sin(60^{\circ})}$ We know the standard trigonometric values: $\sin(30^{\circ}) = \frac{1}{2}$ $\sin(60^{\circ}) = \frac{\sqrt{3}}{2}$ Substitute these values into the ratio: $\frac{R_1}{R_2} = \frac{1/2}{\sqrt{3}/2}$ $\frac{R_1}{R_2} = \frac{1}{2} \times \frac{2}{\sqrt{3}}$ $\frac{R_1}{R_2} = \frac{1}{\sqrt{3}}$ The problem states that the ratio of their ranges is $1:x$. So, we have: $\frac{R_1}{R_2} = \frac{1}{x}$ Comparing this with our calculated ratio: $\frac{1}{x} = \frac{1}{\sqrt{3}}$ Therefore, the value of $x$ is: $x = \sqrt{3}$ This matches option B.
Common Traps & Exam Tip:
1. Missing the $2\theta$ in the formula: A very common mistake is to directly use $\sin(\theta)$ instead of $\sin(2\theta)$ in the range formula. If one were to use $\sin(15^{\circ})$ and $\sin(30^{\circ})$ directly, the result would be incorrect. Always remember that the range formula involves the sine of *twice* the projection angle. 2. Incorrect Trigonometric Values: Errors in recalling or calculating the values of $\sin(30^{\circ})$ or $\sin(60^{\circ})$ can lead to the wrong answer. It's crucial to be proficient with these standard angles. 3. Assumptions about $u$ and $g$: While the problem explicitly states "same initial velocities," sometimes students might accidentally consider them different. Always carefully read the problem statement to identify constants and variables. $g$ is always constant for a given location. 4. Complementary Angles: Remember that the range is the same for projection angles $\theta$ and $(90^\circ - \theta)$ (e.g., $15^\circ$ and $75^\circ$, or $30^\circ$ and $60^\circ$). While this concept is not directly applied here (as $15^\circ$ and $30^\circ$ are not complementary, nor are $30^\circ$ and $60^\circ$), it's a related property of projectile motion that can sometimes lead to confusion if misapplied. Here, $2\theta_1 = 30^\circ$ and $2\theta_2 = 60^\circ$ *are* complementary angles, which means $\sin(2\theta_1) = \sin(90^\circ - 2\theta_2)$, and this is why their sines are directly related ($\sin 30^\circ = \cos 60^\circ$). To avoid these traps, always write down the correct formula, substitute values carefully, and double-check your trigonometric calculations.