How to Solve Projectile Motion: Range, Height & Time of Flight

Solve level-ground projectile motion with T = 2v sinθ/g, R = v² sin(2θ)/g, and H = v² sin²θ/(2g). Worked examples and a free projectile calculator.

What Projectile Motion Asks

A projectile is launched with speed v at angle θ to the horizontal and then moves under gravity alone. Intro physics ignores air resistance and usually lands at the same height it started. Horizontal velocity stays constant. Vertical velocity behaves like one-dimensional free fall.

If you already have v and θ and only need checked T, R, and H, open the free projectile motion calculator first, then return here for the splits and traps.

The Three Formulas (Level Ground)

  • Time of flight: T = 2 v sinθ / g.
  • Range: R = v² sin(2θ) / g.
  • Maximum height: H = v² sin²θ / (2g).

    • Equivalent components: vx = v cosθ, vy = v sinθ. Then T = 2 vy / g, R = vx T, H = vy² / (2g).

    Why 45° Gives Maximum Range

    sin(2θ) peaks at 2θ = 90°, so θ = 45°. Complementary angles such as 30° and 60° share the same sin(2θ) and therefore the same range, but not the same hang time or peak height.

    The vertical part is constant-acceleration kinematics. Refresh SUVAT choice in how to choose the right kinematics equation.

    Units

  • Speed v → m/s (convert km/h first: divide by 3.6).
  • Angle θ → degrees in the calculator; use radians only if your device is in radian mode.
  • g → m/s² (9.8 or 9.81).
  • T → s. R and H → m.

    • Worked Examples

    Example 1: 20 m/s at 45°

    g = 9.8. T ≈ 2.886 s, R ≈ 40.82 m, H ≈ 10.20 m. Peak is at T/2.

    Example 2: 30 m/s at 30°

    T ≈ 3.061 s, R ≈ 79.55 m, H ≈ 11.48 m. A 60° launch with the same speed has the same range and a higher peak.

    Example 3: 10 m/s straight up

    θ = 90°. R = 0, T ≈ 2.041 s, H ≈ 5.10 m.

    When These Formulas Fail

  • Landing on a cliff or a raised platform (Δy ≠ 0).
  • Air resistance, spin, or a strong wind.
  • A moving launch platform unless you include that extra velocity.

    • Common Mistakes

  • Using sinθ in the range formula instead of sin(2θ).
  • Forgetting to convert km/h to m/s.
  • Mixing degree and radian mode.
  • Treating range as the distance along the curve instead of the horizontal landing distance.

    • Check It on the Calculator

    Projectile motion calculator.

    Kinematics calculator.

    Velocity calculator.

    Learn T, R, and H for level ground and check every launch number.

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