JEE PYQ: Motion in a Plane - Question ID bc4ff998510b (JEE Main 2025)

In projectile motion, the maximum height \( H \) attained by a projectile launched with initial speed \( u \) at an angle \( \theta \) with the horizontal is given by:

Here: - \( u \) is the initial speed (same for both projectiles in this problem), - \( \theta \) is the launch angle, - \( g \) is the acceleration due to gravity.

Since both projectiles are fired from the same point with the same initial speed, the ratio of their maximum heights depends only on the ratio of \( \sin^2 \theta \) for their respective angles.

The question gives two angles: - \( \theta_1 = 45^\circ - \alpha \) - \( \theta_2 = 45^\circ + \alpha \)

We are to find the ratio: $\frac{H_1}{H_2} = \frac{\sin^2 (45^\circ - \alpha)}{\sin^2 (45^\circ + \alpha)}$

Step 1: Express \( \sin(45^\circ \pm \alpha) \) using trigonometric identities.

Recall the sine of sum and difference:

Apply this to \( \sin(45^\circ \pm \alpha) \):

Step 2: Square both expressions to get \( \sin^2 \theta \):

Step 3: Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), and the double-angle identity \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \):

Step 4: Compute the ratio \( \frac{H_1}{H_2} \):

This matches option D.

1. Misapplying trigonometric identities: Students often confuse \( \sin^2 \theta \) with \( \sin 2\theta \), or incorrectly expand \( \sin(45^\circ \pm \alpha) \). Always use the sum/difference formulas carefully.

2. Forgetting to square the sine terms: The maximum height depends on \( \sin^2 \theta \), not \( \sin \theta \). A common mistake is to compute the ratio of \( \sin \theta \) instead of \( \sin^2 \theta \).

3. Incorrect simplification: Some students stop at \( (\cos \alpha \pm \sin \alpha)^2 \) and do not simplify using \( \sin 2\alpha \). Always simplify expressions fully to match the given options.

4. Sign errors: When expanding \( (\cos \alpha - \sin \alpha)^2 \), the middle term is negative: \( -2 \sin \alpha \cos \alpha \). A sign error here leads to an incorrect ratio.

Exam Tip: When angles are symmetric around \( 45^\circ \), use trigonometric identities involving \( 45^\circ \) and simplify using double-angle formulas. This often leads to cleaner expressions that match the options.