Question

Two objects are projected with same velocity 'u' at different angles \(α\) and \(β\) with the horizontal. If \(α+β = 90^{\circ}\), the ratio of horizontal range of the first object to the second object will be \([sin(π-θ) = sinθ]\)

• A. \(4:1\)
• B. \(2:1\)
• C. \(1:1\)
• D. \(1:2\)

Hint:

Range depends on sin 2 theta, and sin 2 alpha equals sin 2 beta when alpha plus beta is 90 degrees.

Answer 1

The Correct Option is C

Step 1: Range formula
For a projectile, \(R = \frac{u^2\sin2\theta}{g}\).

Step 2: Complementary angles
With \(\beta = 90^{\circ}-\alpha\), \(2\beta = 180^{\circ}-2\alpha\), so \(\sin2\beta = \sin(180^{\circ}-2\alpha) = \sin2\alpha\).

Step 3: Ratio
\(\frac{R_1}{R_2} = \frac{\sin2\alpha}{\sin2\beta} = 1\). The ratio is \(1:1\), option (C).

Final Answer:
Both ranges are equal, so the ratio is 1:1. \[ \boxed{\text{(C)}\ 1:1} \]