Question

If \(\theta\) is the angle of projection of an object for which the horizontal range is equal to the maximum height attained, then the value of \(\tan \theta\) is

• A. \(\sqrt{2}\)
• B. 1
• C. 4
• D. \(\frac{1}{2}\)
• E. \(\frac{1}{\sqrt{2}}\)

Hint:

\(R = H\) gives \(\tan \theta = 4\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Range \(R = \frac{u^2 \sin 2\theta}{g}\), Maximum height \(H = \frac{u^2 \sin^2 \theta}{2g}\). Set \(R = H\).

Step 2: Detailed Explanation:
\(\frac{u^2 \sin 2\theta}{g} = \frac{u^2 \sin^2 \theta}{2g}\)
\(\sin 2\theta = \frac{\sin^2 \theta}{2}\)
\(2\sin \theta \cos \theta = \frac{\sin^2 \theta}{2}\)
Multiply by 2: \(4\sin \theta \cos \theta = \sin^2 \theta\)
If \(\sin \theta \neq 0\), divide: \(4\cos \theta = \sin \theta\)
\(\tan \theta = 4\)

Step 3: Final Answer:
\(\tan \theta = 4\).