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: Write the horizontal and vertical positions as functions of time: \( x = (u\cos\alpha)t \) and \( y = (u\sin\alpha)t - \dfrac{1}{2}gt^2 \).

Step 2: From the horizontal equation, solve for time in terms of position, \( t = \dfrac{x}{u\cos\alpha} \), then substitute this into the vertical equation to eliminate \( t \) and express \( y \) purely in terms of \( x \).

Step 3: The projectile lands when \( y = 0 \) again (other than at the launch point \( x = 0 \)). Setting the resulting expression to zero and solving for the nonzero value of \( x \) gives \( R = \dfrac{2u^2\sin\alpha\cos\alpha}{g} \), which simplifies using the double-angle identity \( 2\sin\alpha\cos\alpha = \sin 2\alpha \).
\[ \boxed{R = \dfrac{u^2\sin 2\alpha}{g}} \]