Projectile Motion — Horizontal Range on Level Ground For a projectile launched with initial speed u at an elevation angle θ above the horizontal (neglecting air resistance), what is the horizontal range R on level ground?

Introduction / Context:
Horizontal range is a fundamental result in projectile motion used in ballistics, sports, and engineering. On level ground without drag, the closed-form expression depends on the launch speed u, launch angle θ, and gravity g.

Given Data / Assumptions:

• Launch speed u and angle θ.
• Flat ground; same elevation launch and landing.
• No air resistance; constant gravitational acceleration g.

Concept / Approach:
Resolve motion into horizontal and vertical components, determine time of flight from vertical motion, and multiply by horizontal speed to obtain the range. Use the double-angle identity to simplify.

Step-by-Step Solution:

Verification / Alternative check:
Maximum range occurs at θ = 45°, giving R_max = u^2 / g, consistent with differentiating R(θ) for optimum.

Why Other Options Are Wrong:
Option (b) has g^2 in the denominator; (c) incorrectly omits a factor u; (d) uses cos 2θ which is not derived from the horizontal-range calculation.

Common Pitfalls:
Forgetting the same-level assumption; confusing sin 2θ with 2 sin θ or not applying the double-angle identity correctly.

Final Answer:
R = (u^2 * sin 2θ) / g.