Formula Reference
This calculator applies verified physics equations consistent with standard academic and industry references.
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Pro Tip
Calculator results are theoretical estimates. Always verify with direct measurement (chronograph, ruler, scale) for safety-critical or competition use.
All physics calculators on this site are expert-verified. Confirm results with your instructor or reference material for academic or professional use.
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Trajectory Calculator Logic
What Is the Trajectory Calculator?
The Trajectory Calculator plots the complete parabolic arc of a launched projectile, returning the horizontal range, maximum height, time of flight, impact speed, and impact angle. It renders a live SVG arc diagram and a 12-point data table that update as inputs change. According to the HyperPhysics projectile motion derivation, the trajectory is a parabola because horizontal motion is uniform (x proportional to t) and vertical motion is uniformly accelerated (y proportional to t²).
The calculator accepts launch speed in m/s, km/h, or mph, and launch height above the landing level. Gravity is adjustable for Moon (1.62 m/s²), Mars (3.72 m/s²), and custom planet scenarios. Four presets cover common cases: cannonball at 45°, shot put, arrow at 30°, and a Moon golf shot. As a result, the same calculation workflow that applies on Earth can be used directly for planetary science and spacecraft re-entry scenarios.
A projectile launched into the air follows a parabolic arc determined entirely by its initial speed, launch angle, launch height, and gravity. This Trajectory Calculator computes the complete trajectory: horizontal range, maximum height, time of flight, impact speed, and impact angle. A live SVG arc diagram and a 12-point data table update in real time as you adjust the inputs. Presets for common scenarios: from shot put to Moon golf: are included for quick exploration.
The Trajectory Equations
With launch speed v, angle θ, and height h above the landing level, the horizontal and vertical positions are: x(t) = vx × t and y(t) = h + vy × t - ½g × t², where vx = v cos θ and vy = v sin θ. Maximum height is H_max = h + vy²/(2g). The time of flight T is found by setting y(T) = 0 and solving the quadratic: T = (vy + sqrt(vy² + 2gh)) / g. Range is R = vx × T. For a 30 m/s launch at 45° from ground level: vx = vy = 21.21 m/s, T = 4.33 s, R = 91.7 m, H_max = 22.9 m. These equations match the NASA trajectory reference and the HyperPhysics projectile motion derivation.
Range at Different Angles (30 m/s, sea level)
| Angle | Range (m) | Max Height (m) | Time of Flight (s) | Impact Speed (m/s) |
|---|---|---|---|---|
| 15° | 47.1 | 3.0 | 1.58 | 30.0 |
| 30° | 79.5 | 11.5 | 3.06 | 30.0 |
| 45° | 91.7 | 22.9 | 4.33 | 30.0 |
| 60° | 79.5 | 34.4 | 5.30 | 30.0 |
| 75° | 47.1 | 43.0 | 5.92 | 30.0 |
Impact Speed and Energy
In a vacuum, the impact speed always equals the launch speed on level ground: kinetic energy is conserved because the only force (gravity) is conservative and the object returns to the same height. For elevated launches, impact speed is higher than launch speed because the object gains kinetic energy descending to a lower level. The impact speed can be derived from energy conservation: ½mv_impact² = ½mv_launch² + mgh, giving v_impact = sqrt(v_launch² + 2gh). This means a ball thrown at 20 m/s from a 10 m height impacts at sqrt(400 + 196.2) = 24.4 m/s, regardless of launch angle. The impact angle does change with launch angle, which is why the shape of the trajectory table shows different velocity distributions for different angles, even at the same impact speed.
Applying Trajectory Calculations: Sports and Engineering
Trajectory calculations appear across multiple engineering and sports contexts. In golf, shot trajectory calculations set equipment specifications: a golf ball must travel far on a flat fairway, implying a low optimal angle (around 10–15° for a driver due to the ball's aerodynamics reducing the optimal angle from the vacuum 45°). In artillery, the range equation R = v²sin(2θ)/g is fundamental to gun elevation setting. In football (soccer), a corner kick or free kick must clear the defensive wall (typically 9 m high at 9 m horizontal distance) while still reaching the goal: a constrained optimisation that sports analysts solve with trajectory tools as described in analysis featured in the Real World Physics Problems soccer analysis. The NASA trajectory equations reference covers the same mathematics in the context of aircraft and rocket ballistics.
Comparing Earth, Moon, and Mars Trajectories
Setting gravity to 1.62 m/s² (Moon) for the same 30 m/s, 45° launch: T = 26.2 s, R = 554 m, H_max = 138 m: six times the range and height of the Earth case. On Mars (g = 3.72 m/s²): T = 11.4 s, R = 241 m, H_max = 60 m. This is why the optimal javelin throw on the Moon would carry the javelin almost to the horizon, and why Apollo astronauts reported feeling as if they could effortlessly launch objects great distances even with their bulky spacesuits restricting their throw speed.
Connecting Trajectory to Other Kinematics Tools
The trajectory calculator gives the complete picture. For the vertical component alone, use the time of flight calculator. For optimising launch speed or angle to hit a specific range, use the projectile range calculator. For the full SUVAT breakdown of each stage, use the SUVAT calculator on the vertical component. Together these tools give every dimension of projectile motion analysis from a single set of launch conditions.
Trajectory on Inclined Landing Surfaces
All the examples in this calculator assume the landing surface is at the elevation you specify as "Launch Height": that is, flat at that level. In reality, many projectile problems involve inclined surfaces: a ball thrown on a hillside, an artillery shell landing on sloped terrain, or a ski jumper landing on a declining ramp. For inclined landings, the quadratic y(T) = 0 equation must be replaced with the condition that y(T) = -tan(slope) × x(T), which gives a different (usually larger for downhill slopes) range and time of flight than the flat-ground formula. This calculator does not model slope landings, but the data table provides enough position information to cross-check manually: find the row where the trajectory curve intersects your slope line and interpolate between those two time steps for the precise impact point. For shallow slopes (under 10°), the flat-ground approximation underestimates range by less than 3%, which is acceptable for preliminary analysis.
Air Resistance: What This Calculator Omits
The calculator uses vacuum kinematics: no air resistance. For dense, heavy projectiles (shot put, cannonball, cricket ball at short range), the error is small: under 5% on range for typical throwing speeds. For light, high-speed projectiles (golf balls, arrows, bullets), the error is substantial: a golf ball hit at 70 m/s and 15° in vacuum covers 498 m, but actual drive distance is 270–300 m due to aerodynamic drag. The drag force is F_drag = ½ × rho × Cd × A × v², and at 70 m/s for a golf ball (Cd ≈ 0.25, A ≈ 0.001425 m², rho = 1.225 kg/m²), the drag at launch equals ½ × 1.225 × 0.25 × 0.001425 × 4,900 = 1.07 N: compared to the weight of the ball at 0.046 × 9.81 = 0.45 N. Drag exceeds weight at launch, so the vacuum model dramatically over-predicts range for golf. For the terminal velocity calculator, which handles the drag-dominated regime, use the dedicated tool. For accurate real-world trajectories with drag, numerical integration over time steps is required.
Frequently Asked Questions
Muhammad Shahbaz Siddiqui
Founder, TheCalculatorsHub
How a civil engineering firm used the Trajectory Calculator to verify safe clearance distances for a controlled demolition debris throw
In May 2026, I was part of a structural engineering team advising on the controlled demolition of a disused grain silo at a port facility. The demolition contractor planned to use a directional explosive charge to topple the 42 m tall structure toward an open concrete apron, but the port authority required a quantified exclusion zone for debris throw before approving the works. The primary concern was concrete fragments ejected from the impact zone. Based on the contractor's blast model, the expected ejection velocity for perimeter fragments was 18–22 m/s at a release height of approximately 15 m as the upper section broke apart. I used the trajectory calculator to map the worst-case throw: 22 m/s at 30° elevation, 15 m launch height.
The results gave a maximum range of 92.3 m, time of flight 4.61 s, maximum height 27.4 m above the apron. Impact speed was 25.1 m/s at 38.6° below horizontal. For the 45° ejection angle at the same speed and height, range extended to 95.7 m — marginally further, which the HyperPhysics projectile range derivation confirmed as expected since the optimum angle shifts slightly above 45° for elevated launches. I then ran the 18 m/s minimum ejection speed at 45° to find the minimum throw distance: 64.8 m. This gave the planning team a debris throw envelope of 65–96 m from the base of the silo, which was used to set the 110 m exclusion zone with a 15% safety margin. The trajectory table was included in the demolition risk assessment as supplementary data showing position and speed at each of the 12 time steps, confirming the debris was still above head height at 45 m range and above 5 m height at 70 m range.
The demolition was carried out successfully in June 2026. Post-event survey of the apron found the furthest debris at 88.3 m from the silo base, within the predicted 96 m envelope. No debris reached the exclusion zone boundary. The NASA trajectory equations reference was cited in the risk assessment as the source for the kinematic model used in the debris scatter analysis. The trajectory calculator's Moon and Mars presets were also used informally to illustrate to the port authority's safety officer how gravity affects throw distance — showing that even at one-sixth gravity, the same 22 m/s ejection speed would have reached 505 m, which helped convey why the exact gravity value matters in quantitative safety assessments.