JEE PYQ: Motion in a Plane - Question ID a51f2e8c9ee5 (JEE Main 2021)

In projectile motion, the trajectory of a particle launched from the origin with initial velocity $v_0$ at an angle $\theta$ with the horizontal is described by the equation:

This equation is derived by eliminating time from the horizontal and vertical motion equations:

• Horizontal motion: $x = v_0 \cos \theta \cdot t$
• Vertical motion: $y = v_0 \sin \theta \cdot t - \frac{1}{2} g t^2$

The given trajectory equation is $y = \alpha x - \beta x^2$. By comparing this with the standard trajectory equation, we can extract the angle of projection $\theta$ and the maximum height $H$.

Step 1: Compare the given trajectory with the standard form

Given trajectory: $y = \alpha x - \beta x^2$ Standard trajectory: $y = x \tan \theta - \frac{g x^2}{2 v_0^2 \cos^2 \theta}$

By comparing coefficients of $x$ and $x^2$ in both equations, we get:

Step 2: Find the angle of projection $\theta$

From equation (1), we directly obtain: $\theta = \tan^{-1} \alpha$

Step 3: Express $v_0^2$ in terms of $\alpha$ and $\beta$

Recall that $\cos^2 \theta = \frac{1}{1 + \tan^2 \theta} = \frac{1}{1 + \alpha^2}$. Substituting this into equation (2):

Solving for $v_0^2$:

Step 4: Find the maximum height $H$

The maximum height $H$ in projectile motion is given by:

We know $\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}} = \frac{\alpha}{\sqrt{1 + \alpha^2}}$. Thus:

Substituting $v_0^2$ and $\sin^2 \theta$ into the expression for $H$:

Step 5: Match with the given options

From the above derivations:

• Angle of projection: $\theta = \tan^{-1} \alpha$
• Maximum height: $H = \frac{\alpha^2}{4 \beta}$

This matches Option A.

1. Misidentifying coefficients: Students often confuse the coefficients of $x$ and $x^2$ in the trajectory equation, leading to incorrect expressions for $\tan \theta$ or $\beta$. Always compare the given equation with the standard form carefully.

2. Incorrect trigonometric identities: Forgetting that $\cos^2 \theta = \frac{1}{1 + \tan^2 \theta}$ or misapplying $\sin \theta$ in terms of $\tan \theta$ can lead to errors in calculating $H$. Practice deriving these identities to avoid mistakes.

3. Algebraic errors: Simplifying expressions like $v_0^2$ or $H$ requires careful algebra. Double-check each step to ensure no terms are dropped or miscalculated.

4. Unit consistency: While this problem does not involve units explicitly, ensure that all physical quantities are consistent (e.g., $g$ is in $m/s^2$ if $v_0$ is in $m/s$).

Exam Tip: For trajectory-based problems, always start by writing the standard trajectory equation and then compare it with the given equation. This systematic approach minimizes errors and saves time.