Projectile Motion — Cartesian Trajectory Equation Which of the following correctly gives y as a function of x for a projectile launched with speed u at angle α above the horizontal (neglecting air resistance)?

Introduction / Context:
The Cartesian equation of a projectile’s trajectory is a parabola. Deriving y(x) helps solve range, height, and position problems without explicitly using time after elimination.

Given Data / Assumptions:

• Initial speed u, projection angle α above horizontal.
• Gravity g constant; no air resistance.

Concept / Approach:
Resolve initial velocity: u_x = u cos α, u_y = u sin α. Use horizontal motion x = u cos α * t and vertical motion y = u sin α * t − (1/2) g t^2. Eliminate time t by t = x / (u cos α) to get the Cartesian form.

Step-by-Step Derivation:

Verification / Alternative check:
At x = 0 ⇒ y = 0; the curvature term is negative (concave down), consistent with gravity. Units also match length on both sides.

Why Other Options Are Wrong:
The “+” sign before the gravity term would make the path open upward; the other two options are unrelated forms and do not represent y(x) for projectile motion.

Common Pitfalls:
Forgetting the cos^2 α in the denominator; sign errors on the gravity term; failing to eliminate time correctly.

Final Answer:
y = x tan α − (g x^2) / (2 u^2 cos^2 α).