Projectile Motion Experiment Calculator

Enter launch speed, angle, and height to get time of flight, maximum height, horizontal range, and impact speed using standard kinematics equations.

Quick Facts

Time of flight
t = [v₀sinθ + √((v₀sinθ)² + 2gh₀)] / g
Reduces to t = 2v₀sinθ / g when launch height h₀ is zero.
Horizontal range
R = v₀cosθ × t
Horizontal velocity stays constant — no forces act sideways in ideal projectile motion.
Maximum height
H = h₀ + (v₀sinθ)² / (2g)
Measured above the landing surface, reached when vertical velocity hits zero.
Optimal angle
45° maximizes range when h₀ = 0
A launch height above the landing point shifts the optimal angle slightly below 45°.

Your Results

Calculated
Time of Flight
-
Total time in the air, in seconds
Maximum Height
-
Peak height above the landing surface
Horizontal Range
-
Distance traveled before landing
Impact Speed
-
Speed at the moment of landing

Ready

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

Formula and Method for Projectile Motion

Projectile motion describes an object launched into the air and moving under gravity alone, with no thrust and (in the ideal case modeled here) no air resistance. The trajectory separates cleanly into two independent motions: constant-velocity horizontal motion and constant-acceleration vertical motion, both governed by standard kinematics equations. This calculator uses the launch speed v0, launch angle θ, and launch height h0 (measured above the landing surface) to find how long the object stays airborne, how high it climbs, how far it travels, and how fast it is moving when it lands.

How the calculation works

The launch velocity is split into components: horizontal vx = v0 cosθ (stays constant throughout the flight) and vertical vy0 = v0 sinθ (decelerates under gravity g = 9.81 m/s², or 32.174 ft/s²). Setting the vertical position h0 + vy0·t − ½gt² equal to zero and solving the quadratic for t gives the time of flight t = [v0 sinθ + √((v0 sinθ)² + 2gh0)] / g. Horizontal range is then R = vx × t, and maximum height above the landing surface is H = h0 + (v0 sinθ)² / (2g), reached when the vertical velocity momentarily equals zero. Impact speed follows from energy conservation: v_impact = √(v0² + 2gh0), which holds regardless of launch angle.

Common mistakes

  • Entering angle in the wrong units: this calculator expects degrees (0-90), not radians.
  • Forgetting the launch height: a ball rolled off a table or launched from a ramp has h0 > 0, which increases both time of flight and range compared with a ground-level launch at the same speed and angle.
  • Mixing max height above launch point with max height above the ground: this calculator reports height above the landing surface (h0 plus the climb), not height gained above the launch point.
  • Ignoring air resistance for light or fast projectiles: ping-pong balls, badminton shuttles, and anything moving fast relative to its mass will land short of the ideal (drag-free) prediction.

Real-world applications

  • Introductory physics labs use tabletop ramps or spring launchers to measure range and compare it against the predicted formula.
  • Sports science applies these equations to estimate the range of a thrown or kicked ball from its launch speed and angle.
  • Ballistics and artillery calculations use the same kinematics as a starting point before adding drag and wind corrections.
  • Engineering tests (nozzle sprays, water jets, ejected debris) use projectile range to back-calculate an unknown launch speed.

Frequently Asked Questions

What is the time-of-flight formula for projectile motion?
For a projectile launched with speed v0 at angle θ from height h0 above the landing surface, time of flight is t = [v0 sinθ + √((v0 sinθ)² + 2 g h0)] / g. If the launch height is zero, this simplifies to t = 2 v0 sinθ / g.
How is the maximum height of a projectile calculated?
Maximum height above the ground is H = h0 + (v0 sinθ)² / (2g), where h0 is the launch height, v0 is the launch speed, θ is the launch angle above horizontal, and g is gravitational acceleration (9.81 m/s² on Earth). The projectile reaches this peak when its vertical velocity momentarily equals zero.
Does this calculator include air resistance?
No. Like standard introductory-physics treatments, this calculator assumes ideal projectile motion: constant gravitational acceleration and no air drag. Light or fast-moving objects (a ping-pong ball, a badminton shuttle) can land noticeably short of the ideal range because of drag.
What launch angle gives the maximum range?
When the launch height is zero, range is maximized at a 45° launch angle. If the projectile launches from above the landing height (h0 > 0), the optimal angle is slightly less than 45° — the extra fall time favors a flatter trajectory that trades height for horizontal speed.

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