Question

Two projectiles are thrown at angles $ \theta $ and $ (90^\circ - \theta) $ with same initial velocity and range $ R $. If maximum height attained by them are $ H_1 $ and $ H_2 $ respectively, then

• A. \( R = H_1 + H_2 \)
• B. \( R = \sqrt{H_1^2 + H_2^2} \)
• C. \( R = \sqrt{H_1 H_2} \)
• D. \( R = 4 \sqrt{H_1 H_2} \)

Hint:

The range and height of projectiles are related through their initial velocity and launch angles. For projectiles launched at complementary angles, their range is the geometric mean of their individual heights.

Answer 1

The Correct Option is C

Step 1: Use the relationship between the maximum height and range for projectile motion.
The maximum height for a projectile is
given by:
\[ H = \frac{v^2 \sin^2 \theta}{2g} \] where \( v \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.
Step 2: Analyze the two projectiles.
For the two projectiles: \( H_1 = \frac{v^2 \sin^2 \theta}{2g} \) \( H_2 = \frac{v^2 \cos^2 \theta}{2g} \) The range of a projectile is
given by:
\[ R = \frac{v^2 \sin(2\theta)}{g} \]
Step 3: Conclusion.
The relationship between \( R \) and \( H_1, H_2 \) is \( R = \sqrt{H_1 H_2} \).
Conclusion:
The correct answer is (C) \( R = \sqrt{H_1 H_2} \).