{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"projectile-motion"},"result":{"entity_type":"calculator","id":"projectile-motion","calculator_id":"projectile-motion","canonical_url":"https://tttkmbb.com/physics/projectile-motion","name":"Projectile Motion Calculator","title":"Projectile Motion Calculator – Range, Maximum Height and Time of Flight with Launch Height","category":"physics","category_name":"Physics","tool_name":"calculate_projectile_motion","featured_mcp_tool":false,"description":"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).","use_when":"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).","inputs":[{"name":"initial_speed_m_s","label":"Launch speed","type":"number","unit":"m/s","required":true,"min":0,"description":"Initial speed in metres per second.","example":20},{"name":"angle_degrees","label":"Launch angle","type":"number","unit":"°","required":true,"min":0,"max":90,"description":"Angle above the horizontal in degrees (0 = horizontal, 90 = straight up).","example":45},{"name":"initial_height_m","label":"Launch height","type":"number","unit":"m","required":false,"default":0,"min":0,"description":"Height of the launch point above the landing level, in metres.","example":0},{"name":"gravity_m_s2","label":"Gravitational acceleration","type":"number","unit":"m/s²","required":false,"default":9.80665,"max":10000,"exclusive_min":0,"description":"Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8).","example":9.80665}],"outputs":[{"name":"time_of_flight_s","label":"Time of flight","type":"number","unit":"s","decimals":4,"description":"Time until the projectile returns to the landing level."},{"name":"range_m","label":"Horizontal range","type":"number","unit":"m","decimals":4,"description":"Horizontal distance travelled at landing."},{"name":"max_height_m","label":"Maximum height","type":"number","unit":"m","decimals":4,"description":"Peak height above the landing level (includes the launch height)."},{"name":"time_to_max_height_s","label":"Time to peak","type":"number","unit":"s","decimals":4,"description":"Time at which the vertical velocity is zero: v·sin θ / g."},{"name":"horizontal_velocity_m_s","label":"Horizontal velocity","type":"number","unit":"m/s","decimals":4,"description":"Constant horizontal component v·cos θ."},{"name":"initial_vertical_velocity_m_s","label":"Initial vertical velocity","type":"number","unit":"m/s","decimals":4,"description":"Vertical component at launch v·sin θ."},{"name":"impact_speed_m_s","label":"Impact speed","type":"number","unit":"m/s","decimals":4,"description":"Speed at landing: √(v² + 2·g·h0)."}],"input_schema":{"type":"object","properties":{"initial_speed_m_s":{"description":"Initial speed in metres per second. Unit: m/s.","type":"number","minimum":0,"examples":[20],"x-unit":"m/s"},"angle_degrees":{"description":"Angle above the horizontal in degrees (0 = horizontal, 90 = straight up). Unit: °.","type":"number","minimum":0,"maximum":90,"examples":[45],"x-unit":"°"},"initial_height_m":{"description":"Height of the launch point above the landing level, in metres. Unit: m.","type":"number","minimum":0,"default":0,"examples":[0],"x-unit":"m"},"gravity_m_s2":{"description":"Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8). Unit: m/s².","type":"number","maximum":10000,"exclusiveMinimum":0,"default":9.80665,"examples":[9.80665],"x-unit":"m/s²"}},"additionalProperties":false,"required":["initial_speed_m_s","angle_degrees"]},"output_schema":{"type":"object","properties":{"time_of_flight_s":{"description":"Time until the projectile returns to the landing level. Unit: s.","type":"number","x-unit":"s"},"range_m":{"description":"Horizontal distance travelled at landing. Unit: m.","type":"number","x-unit":"m"},"max_height_m":{"description":"Peak height above the landing level (includes the launch height). Unit: m.","type":"number","x-unit":"m"},"time_to_max_height_s":{"description":"Time at which the vertical velocity is zero: v·sin θ / g. Unit: s.","type":"number","x-unit":"s"},"horizontal_velocity_m_s":{"description":"Constant horizontal component v·cos θ. Unit: m/s.","type":"number","x-unit":"m/s"},"initial_vertical_velocity_m_s":{"description":"Vertical component at launch v·sin θ. Unit: m/s.","type":"number","x-unit":"m/s"},"impact_speed_m_s":{"description":"Speed at landing: √(v² + 2·g·h0). Unit: m/s.","type":"number","x-unit":"m/s"}}},"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)","method":"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.","sources":[{"name":"HyperPhysics – Trajectories","url":"http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Projectile motion","url":"https://en.wikipedia.org/wiki/Projectile_motion","type":"reference","retrieved_at":"2026-09-23"},{"name":"NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s²","url":"https://physics.nist.gov/cgi-bin/cuu/Value?gn","type":"standard","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"20 m/s at 45° from ground level","inputs":{"initial_speed_m_s":20,"angle_degrees":45},"expected":{"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},"url":"https://tttkmbb.com/api/v1/calculate/projectile-motion?initial_speed_m_s=20&angle_degrees=45"},{"name":"30 m/s at 30° from 10 m height","inputs":{"initial_speed_m_s":30,"angle_degrees":30,"initial_height_m":10},"expected":{"time_of_flight_s":3.6222,"range_m":94.1072,"max_height_m":21.4718,"impact_speed_m_s":33.1079},"url":"https://tttkmbb.com/api/v1/calculate/projectile-motion?initial_speed_m_s=30&angle_degrees=30&initial_height_m=10"}],"faq":[{"q":"Which angle gives the maximum range?","a":"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."},{"q":"How realistic is the no-drag result?","a":"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."}],"tags":["projectile motion","range","trajectory","maximum height","time of flight","launch angle"],"related":[{"calculator_id":"free-fall","reason":"The special case of a dropped object."},{"calculator_id":"quadratic-equation","reason":"Solve the flight-time quadratic for other landing heights."},{"calculator_id":"speed-distance-time","reason":"Horizontal motion is uniform."}],"links":{"html":"https://tttkmbb.com/physics/projectile-motion","markdown":"https://tttkmbb.com/physics/projectile-motion.md","json":"https://tttkmbb.com/physics/projectile-motion.json","api":"https://tttkmbb.com/api/v1/calculate/projectile-motion","schema":"https://tttkmbb.com/api/v1/calculators/projectile-motion","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:24:36Z"}