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Projectile Motion Calculator

Computes time of flight, horizontal range, maximum height and impact speed for a projectile launched at a given speed and angle from an optional initial height, ignoring air resistance (closed-form kinematics).

When to use

You need where and when a thrown, kicked or fired object lands on level ground, or its peak height, from launch speed and angle (and launch height).

Do not use when: Air drag is significant (golf balls, bullets, shuttlecocks), the landing point is higher than the launch point, or the object is simply dropped (use free-fall).

Formula

vx = v·cos θ, vy = v·sin θ; time_of_flight = (vy + √(vy² + 2·g·h0)) / g; range = vx × time_of_flight; max_height = h0 + vy² / (2g); time_to_max_height = vy / g; impact_speed = √(v² + 2·g·h0)

Point projectile, no drag, uniform g, level landing surface; the time of flight is the positive root of h0 + vy·t − ½·g·t² = 0.

Inputs

Parameter	Type	Unit	Required	Description
`initial_speed_m_s`	number	m/s	yes	Initial speed in metres per second. Range: ≥ 0
`angle_degrees`	number	°	yes	Angle above the horizontal in degrees (0 = horizontal, 90 = straight up). Range: ≥ 0, ≤ 90
`initial_height_m`	number	m	default 0	Height of the launch point above the landing level, in metres. Range: ≥ 0
`gravity_m_s2`	number	m/s²	default 9.80665	Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8). Range: > 0, ≤ 10000

Outputs

Output	Type	Unit	Description
`time_of_flight_s`	number	s	Time until the projectile returns to the landing level.
`range_m`	number	m	Horizontal distance travelled at landing.
`max_height_m`	number	m	Peak height above the landing level (includes the launch height).
`time_to_max_height_s`	number	s	Time at which the vertical velocity is zero: v·sin θ / g.
`horizontal_velocity_m_s`	number	m/s	Constant horizontal component v·cos θ.
`initial_vertical_velocity_m_s`	number	m/s	Vertical component at launch v·sin θ.
`impact_speed_m_s`	number	m/s	Speed at landing: √(v² + 2·g·h0).

Example

20 m/s at 45° from ground level: {"initial_speed_m_s":20,"angle_degrees":45} → {"time_of_flight_s":2.8842,"range_m":40.7886,"max_height_m":10.1972,"time_to_max_height_s":1.4421,"horizontal_velocity_m_s":14.1421,"impact_speed_m_s":20}

30 m/s at 30° from 10 m height: {"initial_speed_m_s":30,"angle_degrees":30,"initial_height_m":10} → {"time_of_flight_s":3.6222,"range_m":94.1072,"max_height_m":21.4718,"impact_speed_m_s":33.1079}

GET https://tttkmbb.com/api/v1/calculate/projectile-motion?initial_speed_m_s=20&angle_degrees=45

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/projectile-motion (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/projectile-motion · Markdown: https://tttkmbb.com/physics/projectile-motion.md · JSON definition: https://tttkmbb.com/physics/projectile-motion.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="projectile-motion"
OpenAPI operationId: calculate_projectile_motion
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

HyperPhysics – Trajectories (reference)
Wikipedia – Projectile motion (reference)
NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s² (standard)

FAQ

Which angle gives the maximum range?

45° when launching from ground level; the optimum is lower than 45° when launching from a height and higher when the target is above the launch point.

How realistic is the no-drag result?

Good for dense, slow objects (shot put, thrown stones). For a golf ball or bullet the real range can be far shorter than the calculated value.

Related calculators

Free Fall Calculator — The special case of a dropped object.
Quadratic Equation Solver — Solve the flight-time quadratic for other landing heights.
Speed, Distance & Time Calculator — Horizontal motion is uniform.