Question

Four athletes A, B, C and D in a long jump competition each running at a speed of 30 km h$^{-1}$. While jumping, they make angles of 30°, 45°, 60° and 75° with the ground respectively. Then the winner of the competition is:

• A. A
• B. B
• C. C
• D. D

Hint:

For maximum range in projectile motion, angle should be 45° because \(\sin 2\theta\) becomes maximum at 90°.

Answer 1

The Correct Option is B

Step 1: Understanding the physics of long jump.
In projectile motion (long jump), the horizontal range is given by: \[ R = \frac{u^2 \sin 2\theta}{g} \] Here, all athletes have same speed \(u = 30\) km/h, so range depends only on \(\sin 2\theta\).

Step 2: Convert speeds comparison into angle comparison.
Since speed is same for all athletes, we only compare: \[ R \propto \sin 2\theta \] Thus, the athlete with maximum \(\sin 2\theta\) will win.

Step 3: Compute values of \(\sin 2\theta\).
- A: \(2\theta = 60^\circ \Rightarrow \sin 60^\circ = \frac{\sqrt{3}}{2}\)
- B: \(2\theta = 90^\circ \Rightarrow \sin 90^\circ = 1\)
- C: \(2\theta = 120^\circ \Rightarrow \sin 120^\circ = \frac{\sqrt{3}}{2}\)
- D: \(2\theta = 150^\circ \Rightarrow \sin 150^\circ = \frac{1}{2}\)

Step 4: Compare all ranges.
Maximum value is: \[ \sin 90^\circ = 1 \] So athlete B has maximum range.

Step 5: Final conclusion.
Hence, the winner of the competition is: \[ \boxed{B} \]