Trajectory Calculator

Enter a projectile's launch velocity, angle, and height to calculate its time of flight, maximum height, horizontal range, and impact speed using standard kinematic equations.

Quick Facts

Range formula
R = v0² sin(2θ) / g
Applies when launch and landing heights are equal; range peaks at a 45° launch angle.
Time of flight
t = [v0 sinθ + √((v0 sinθ)² + 2gh0)] / g
Reduces to t = 2v0 sinθ / g when the launch height h0 is zero.
Maximum height
H = h0 + (v0 sinθ)² / (2g)
Only the vertical velocity component (v0 sinθ) determines how high the projectile rises.
Standard gravity
g = 9.81 m/s² (32.2 ft/s²) on Earth
Use 1.62 m/s² for the Moon or 3.71 m/s² for Mars.

Your Results

Calculated
Time of Flight
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Total time in the air, in seconds
Maximum Height
-
Peak height above the ground
Horizontal Range
-
Downrange distance at landing
Impact Speed
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Speed at the moment of landing

Ready

Enter launch velocity, angle, and height, then press Calculate.

Formula and Method for Projectile Motion Trajectory

Projectile motion describes the curved path — a parabola, in the absence of air resistance — traced by an object launched into the air and acted on only by gravity. The motion splits cleanly into two independent components: a constant horizontal velocity and a uniformly accelerated vertical velocity. That separation is what lets this calculator solve directly for flight time, maximum height, horizontal range, and landing speed from the launch conditions.

How the calculation works

The launch velocity v0 is split into horizontal and vertical components using the launch angle θ: vx = v0 cos θ and vy = v0 sin θ. Because gravity acts only vertically, vx stays constant for the entire flight while vy decreases (then reverses) at a constant rate g. Solving the vertical position equation h0 + vy·t − ½gt² = 0 for the moment the projectile reaches the ground gives the time of flight, t = [v0 sinθ + √((v0 sinθ)² + 2gh0)] / g. Multiplying the constant horizontal speed vx by that time gives the horizontal range, R = vx·t. Maximum height is reached when the vertical velocity hits zero, at H = h0 + (v0 sinθ)² / (2g). Impact speed follows directly from energy conservation — ½v0² + gh0 = ½v_impact² — so v_impact = √(v0² + 2gh0).

Common mistakes and modeling limits

  • Degrees vs. radians: the launch angle is entered in degrees here; if you check the math by hand in code, remember that sin() and cos() functions typically expect radians (multiply degrees by π/180 first).
  • Forgetting launch height: leaving h0 at 0 when a projectile is actually launched from a platform, cliff, or elevated release point understates both flight time and range.
  • No air resistance: this calculator uses ideal, drag-free kinematics. Real projectiles — especially light or slow-moving ones — lose range and height to air drag, and spinning projectiles curve further due to the Magnus effect.
  • 45° isn't always optimal: a 45° launch angle only maximizes range when launch and landing heights are equal. Launching from above the landing point, the optimal angle is slightly below 45°.

Real-world applications

  • Sports: estimating the range of a thrown ball, golf shot, or javelin throw from launch speed and angle.
  • Ballistics: estimating flight time and impact point for a known muzzle velocity and elevation angle as a first-pass, drag-free estimate.
  • Engineering and safety: sizing catch nets, clearance heights, or exclusion zones around launch or ejection equipment.
  • Physics education: verifying the classic results that range is maximized at 45° and that flight time depends only on the vertical velocity component.

Frequently Asked Questions

What is the formula for projectile range?
For a projectile launched and landing at the same height, horizontal range is R = v0² sin(2θ) / g, where v0 is the initial speed, θ is the launch angle, and g is gravitational acceleration. Range is greatest at a 45° launch angle when launch and landing heights match.
How do you find the maximum height of a projectile?
Maximum height above the ground is H = h0 + (v0 sinθ)² / (2g), where h0 is the launch height. Only the vertical component of the initial velocity (v0 sinθ) determines how high the projectile rises.
Does launch height affect time of flight?
Yes. Launching from a height above the landing point increases flight time because the projectile has farther to fall. The full formula is t = [v0 sinθ + √((v0 sinθ)² + 2gh0)] / g, which reduces to t = 2v0 sinθ / g when h0 = 0.
Does this calculator include air resistance?
No. Like most textbook projectile-motion calculations, this tool assumes ideal conditions with no air resistance, spin, or wind. Real-world drag reduces range and height, especially for lightweight or high-speed projectiles.

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