How to approach projectile motion problems — horizontal vs vertical decomposition

Question

A ball is thrown at $20$ m/s at an angle of $60°$ above the horizontal from ground level. Find (a) the time of flight, (b) the maximum height, and (c) the horizontal range. Take $g = 10$ m/s$^2$.

(JEE Main & NEET standard problem)

Solution — Step by Step

Why This Works

Gravity only acts vertically. This means horizontal velocity never changes — the ball moves at constant speed horizontally. Vertically, it’s just a ball thrown straight up and coming back down.

By decomposing into two independent 1D problems, we turn a curved path into two straight-line motions. Time is the link between them — whatever time the vertical motion gives us, the horizontal motion uses the same time.

graph TD
    A["Projectile Problem"] --> B["Decompose velocity"]
    B --> C["u_x = u cos θ"]
    B --> D["u_y = u sin θ"]
    C --> E["Horizontal: constant velocity<br/>x = u_x × t"]
    D --> F["Vertical: uniformly accelerated<br/>y = u_y t - ½gt²"]
    F --> G["Time of flight<br/>T = 2u_y / g"]
    F --> H["Max height<br/>H = u_y² / 2g"]
    G --> I["Range = u_x × T"]

Alternative Method — Using the Trajectory Equation

For problems where you need height at a specific horizontal distance $x$:

This eliminates time entirely. Useful when the question gives $x$ and asks for $y$, or vice versa.

Common Mistake