Question

The horizontal range of a projectile (R) is given by: ____.

• A. R = u² cos 2$\alpha$ g
• B. R = u² sin 2$\alpha$ g
• C. R = u² cos $\alpha$ g
• D. R = u² sin $\alpha$ g

Hint:

The range is maximum when $\sin 2\alpha = 1$, which happens at an angle of projection of $45^\circ$. Also, the range is the same for complementary angles (e.g., $30^\circ$ and $60^\circ$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The horizontal range ($R$) is the total horizontal distance traveled by the projectile from its point of launch to the point where it returns to the same horizontal level.

Step 2: Key Formula or Approach:
Range is the product of the constant horizontal velocity ($u \cos \alpha$) and the total time of flight ($T = 2u \sin \alpha / g$).

Step 3: Detailed Explanation:
Calculation of Range ($R$): \[ R = (u \cos \alpha) \times \left( \frac{2u \sin \alpha}{g} \right) \] \[ R = \frac{u^2 (2 \sin \alpha \cos \alpha)}{g} \] Using the trigonometric double-angle identity ($2 \sin \alpha \cos \alpha = \sin 2\alpha$): \[ R = \frac{u^2 \sin 2\alpha}{g} \]

Step 4: Final Answer:
The horizontal range is given by $R = \frac{u^2 \sin 2\alpha}{g}$.