Projectile motion: horizontal range formula For a projectile launched with initial speed u at an angle α to the horizontal (neglecting air resistance), which expression gives the horizontal range R on level ground?

Introduction / Context:
Projectile range on level ground is among the most common kinematics results. It relates launch speed and angle to the horizontal distance traveled, assuming uniform gravity and no air drag.

Given Data / Assumptions:

• Initial speed u, projection angle α.
• Flat landing at the same elevation as launch.
• Constant gravitational acceleration g, no air resistance.

Concept / Approach:

Resolve velocity into horizontal and vertical components and use independent motions. Time of flight depends on vertical motion; horizontal range equals horizontal speed multiplied by time of flight.

Step-by-Step Solution:

Verification / Alternative check:

Maximum range occurs at α = 45°, giving R_max = u^2 / g, which matches the formula with sin 90° = 1.

Why Other Options Are Wrong:

(b) is equivalent to (a) but lacks the identity recognition; only one should be selected. (c) and (d) are incomplete component squares. (e) is dimensionally incorrect (missing u factor).

Common Pitfalls:

Forgetting the double-angle identity; applying the level-ground formula to unequal elevations without modification.

Final Answer:

R = (u^2 sin 2α) / g