JEE PYQ: Motion in a Plane - Question ID 835c5f2fb929 (JEE Main 2022)

In projectile motion, two key quantities are the range ($R$) and the maximum height ($h$).

• Range formula: $R = \frac{u^2 \sin 2\theta}{g}$ where $u$ is the initial speed, $\theta$ is the launch angle, and $g$ is the acceleration due to gravity.
• Maximum height formula: $h = \frac{u^2 \sin^2 \theta}{2g}$
• Complementary angles property: Two angles $\theta$ and $(90^\circ - \theta)$ yield the same range because $\sin 2\theta = \sin 2(90^\circ - \theta)$.

Step 1: Identify the two angles
Let ball A be launched at angle $\theta$ and ball B at angle $(90^\circ - \theta)$. Both have the same initial speed $u$ and therefore the same range $R$.

Step 2: Express the two heights
For ball A: $h_1 = \frac{u^2 \sin^2 \theta}{2g}$ For ball B: $h_2 = \frac{u^2 \sin^2 (90^\circ - \theta)}{2g} = \frac{u^2 \cos^2 \theta}{2g}$

Step 3: Compute the product $h_1 h_2$
Multiply the two heights: $h_1 h_2 = \Bigl(\frac{u^2 \sin^2 \theta}{2g}\Bigr)\Bigl(\frac{u^2 \cos^2 \theta}{2g}\Bigr) = \frac{u^4 \sin^2 \theta \cos^2 \theta}{4g^2}$ Use the double-angle identity $\sin 2\theta = 2\sin\theta\cos\theta$ to rewrite: $\sin^2\theta\cos^2\theta = \frac{\sin^2 2\theta}{4}$ Hence $h_1 h_2 = \frac{u^4}{4g^2}\;\frac{\sin^2 2\theta}{4} = \frac{u^4 \sin^2 2\theta}{16g^2}$

Step 4: Express $R$ in terms of $u$ and $\theta$
From the range formula: $R = \frac{u^2 \sin 2\theta}{g}$ Square both sides: $R^2 = \frac{u^4 \sin^2 2\theta}{g^2}$

Step 5: Relate $R^2$ to $h_1 h_2$
Compare the expressions: $R^2 = \frac{u^4 \sin^2 2\theta}{g^2} = 16\;\frac{u^4 \sin^2 2\theta}{16g^2} = 16\,(h_1 h_2)$ Take the square root: $R = 4\sqrt{h_1 h_2}$

Step 6: Verify Reason R
Reason R states: $h_1 h_2 = \Bigl(\frac{u^2 \sin^2 \theta}{2g}\Bigr)\Bigl(\frac{u^2 \cos^2 \theta}{2g}\Bigr)$ This is exactly the product we computed in Step 3. Thus Reason R is a correct statement and it directly explains how the product of heights leads to the relation in Assertion A.

Both Assertion A and Reason R are true, and Reason R provides the correct explanation for Assertion A.

• Forgetting complementary angles: Students often miss that two angles adding to $90^\circ$ give the same range. Without this, they cannot pair $h_1$ and $h_2$ correctly.
• Misapplying trigonometric identities: A common error is not using $\sin 2\theta = 2\sin\theta\cos\theta$ to simplify $\sin^2\theta\cos^2\theta$.
• Algebraic slip in squaring: When relating $R^2$ to $h_1h_2$, one must carefully track the factor of $16$ to avoid an incorrect numerical coefficient.
• Exam tip: Always check if the Reason actually derives the Assertion. Here, multiplying the two heights and simplifying via $\sin 2\theta$ directly yields $R=4\sqrt{h_1h_2}$.