Time of Flight Formula: Projectile Motion (T, Range & Height)

Find time of flight with T = 2v sinθ/g, then range and max height. Worked examples plus a free projectile motion calculator for level-ground launches.

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 time-of-flight steps and traps.

Time of Flight Formula

On level ground, time of flight (hang time) is T = 2 v sinθ / g. With components vx = v cosθ and vy = v sinθ, that is the same as T = 2 vy / g — twice the time to the peak.

  • Time to peak: t_up = vy / g = v sinθ / g.
  • Full flight (same height): T = 2 t_up = 2 v sinθ / g.
  • At the peak, vertical velocity is zero; horizontal velocity is still vx.

    • If the landing is higher or lower than the launch (a cliff or raised platform), do not use this T. Solve Δy = vy t − ½ g t² for t, then x = vx t.

    How to Solve for Time in Projectile Motion

  • Convert speed to m/s and keep θ in degrees for the calculator (or use radian mode consistently by hand).
  • Find vy = v sinθ.
  • If launch height = landing height, T = 2 vy / g.
  • Peak time is T/2; that is when height is maximum.

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

    Range and Maximum Height

  • Range: R = v² sin(2θ) / g = vx T.
  • Maximum height: H = v² sin²θ / (2g) = vy² / (2g).

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

    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 (Time of Flight First)

    Example 1: Find T for 20 m/s at 45°

    g = 9.8. vy = 20 sin45° ≈ 14.14 m/s. T = 2(14.14) / 9.8 ≈ 2.886 s. Peak at ≈ 1.443 s. Then R ≈ 40.82 m and H ≈ 10.20 m.

    Example 2: Find time for 30 m/s at 30°

    vy = 30 × 0.5 = 15 m/s. T = 2(15) / 9.8 ≈ 3.061 s. R ≈ 79.55 m, H ≈ 11.48 m. Same speed at 60° keeps R but raises T and H.

    Example 3: 10 m/s straight up

    θ = 90°. R = 0, T = 2(10)/9.8 ≈ 2.041 s, H ≈ 5.10 m — same as 1D free-fall up and down.

    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.
  • Using T = 2v sinθ/g when the ball lands at a different height.

    • Check It on the Calculator

    Projectile motion calculator.

    Kinematics calculator.

    Velocity calculator.

    Learn time of flight first, then check R and H for every level-ground launch.

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