Projectile Motion Basics for Physics Homework

Projectile motion describes the path of an object launched with initial velocity and moving under gravity alone, with air resistance neglected. The key insight is independence of motion: horizontal velocity stays constant (no horizontal force), while vertical motion is uniformly accelerated at g ≈ 9.8 m/s² downward.

Decomposing initial velocity

Launch speed v₀ at angle θ above horizontal splits into components: v₀ₓ = v₀ cos θ (horizontal) and v₀ᵧ = v₀ sin θ (vertical). These components enter separate kinematic equations. At the peak, vertical velocity is zero but horizontal velocity remains v₀ₓ.

Key quantities to calculate

• Time of flight — when vertical displacement returns to launch height
• Maximum height — when vᵧ = 0, using v² = v₀² + 2aΔy
• Horizontal range — R = v₀ₓ × total flight time
• Velocity at any point — combine vₓ and vᵧ components

Symmetry at same launch and landing height

When launch and landing elevations are equal, the ascent and descent times are equal. The speed at landing equals the launch speed (if air resistance is ignored), though the velocity direction differs. The trajectory is symmetric about the peak.

Maximum range angle

For level ground, range R = (v₀² sin 2θ) / g is maximized when sin 2θ = 1, i.e. θ = 45°. Complementary angles (e.g. 30° and 60°) give the same range on level ground with the same launch speed.

Problem-solving strategy

Draw a diagram, choose a coordinate system, list known and unknown variables separately for x and y, pick kinematic equations with one unknown each, and solve. The Projectile Motion Calculator handles the algebra for range, height, and flight time given launch speed and angle.

Frequently asked questions