Compute the full trajectory, range, and maximum height of a projectile
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Faiq Ur Rahman is a web designer, digital product developer, and founder of Toolraxy, a growing platform of web-based calculators and utility tools. He specializes in building structured, user-friendly tools focused on health, finance, productivity, and everyday problem-solving.
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There’s something almost magical about watching a projectile arc through the air, the graceful curve that connects launch to landing, governed by nothing more than gravity and a bit of velocity. Whether you’re designing a basketball shot, analyzing artillery trajectories, or simply exploring the physics of motion, understanding the full flight path tells you far more than just the landing point.
This trajectory calculator computes the complete projectile path, showing you the range, maximum height, time of flight, impact speed, and impact angle for any launch condition. Unlike basic range calculators, this tool visualizes the entire parabolic curve on an interactive chart, helping you see exactly how the projectile moves through space. Adjust launch speed, angle, height, and even gravity switch from Earth to Mars or the Moon with a click. All calculations run locally in your browser, so your data stays private while you explore the fascinating physics of projectile motion.
Enter the initial launch speed and choose your preferred unit from m/s, km/h, mph, or ft/s.
Specify the launch angle in degrees above horizontal, 0° launches horizontally, 45° gives maximum range on level ground.
Set the starting height if the projectile launches from an elevated position like a cliff or platform.
Select the gravitational environment, choose from Earth, Moon, Mars, or Jupiter presets, or enter a custom gravity value.
Click Calculate or simply wait — results update automatically with full trajectory data and an interactive chart.
Review the range, maximum height, time of flight, impact speed, and impact angle in the results section.
Use the Copy button to save a complete summary or Share to send results to colleagues or classmates.
The calculator models projectile motion using the standard parametric equations that describe an object moving under constant gravitational acceleration, with no air resistance. The horizontal position and vertical position are calculated independently.
Formula: x(t) = v₀·cos(θ)·t (horizontal position at time t)
Formula: y(t) = h₀ + v₀·sin(θ)·t − ½·g·t² (vertical position at time t)
Where x is horizontal displacement, y is vertical height, v₀ is initial speed, θ is launch angle, h₀ is initial height, g is gravitational acceleration, and t is elapsed time. The trajectory forms a parabola in the x-y plane.
The calculator determines the time of flight by solving for t when y(t) = 0. For ground-level launches, this simplifies to t = 2·v₀·sin(θ)/g. With elevation, the quadratic solution handles the more general case. Range is simply x(t) at the landing time. Maximum height occurs when vertical velocity reaches zero, at time t_peak = v₀·sin(θ)/g.
Formula: H_max = h₀ + (v₀²·sin²θ)/(2g)
Impact speed combines the constant horizontal velocity with the vertical velocity at landing:
Formula: v_impact = √((v₀·cosθ)² + (v₀·sinθ − g·t_total)²)
The calculator automatically converts all input units to SI before computation, then displays outputs in multiple unit systems for your convenience.
Imagine firing a projectile from the top of a 50-meter cliff at 60 m/s, angled 30° above horizontal. Using Earth’s gravity (9.81 m/s²), what does the full trajectory look like?
Step 1: Break velocity into components
v_x = 60·cos(30°) = 60×0.866 = 51.96 m/s
v_y = 60·sin(30°) = 60×0.5 = 30.0 m/s
Step 2: Find time of flight
Solve the quadratic: 0 = 50 + 30·t − 0.5×9.81·t²
0 = 50 + 30t − 4.905t²
Using the quadratic formula, t_total ≈ 7.87 seconds
Step 3: Calculate the range
Range = v_x·t_total = 51.96 × 7.87 ≈ 409 meters
Step 4: Find maximum height
H_max = 50 + (60²·sin²30°)/(2×9.81) = 50 + (3600×0.25)/(19.62) = 50 + 45.87 = 95.87 meters
Step 5: Impact speed and angle
v_y_impact = 30 − 9.81×7.87 = 30 − 77.2 = −47.2 m/s
v_impact = √(51.96² + 47.2²) = √(2700 + 2228) = √4928 ≈ 70.2 m/s
Impact angle = arctan(|−47.2|/51.96) ≈ 42.3° below horizontal
Interpretation: The projectile reaches a maximum height of 95.9 meters (45.9 meters above the launch point) after 3.06 seconds, travels 409 meters horizontally, and impacts the ground at about 70 m/s at an angle of 42° below the horizontal. The trajectory chart shows the beautiful parabolic arc from cliff top to landing
A firefighter needs to aim a water hose at a burning building. The nozzle is 10 meters above ground, and water exits at 30 m/s. The building is 40 meters away and the fire is on the third floor, 15 meters above ground. What angle should the firefighter use?
Step 1: Use the trajectory equation
The water must reach x = 40 m and y = 15 m at the same time t. We have two equations: x = 30·cosθ·t and y = 10 + 30·sinθ·t − 4.905·t².
Step 2: Solve for θ
Using the range formula for elevated launch, the calculator gives approximately 38° for the shallow trajectory and 52° for the steep trajectory.
Step 3: Choose the practical solution
The 38° trajectory is more direct and gives the water a flatter path, making it easier to aim. The 52° trajectory would also work but requires more elevation.
Interpretation: The firefighter should aim the hose at about 38° above horizontal, using the calculator to verify the trajectory clears the building and reaches the target window accurately.
Visualizes the complete flight path with an interactive chart, showing the parabolic arc in real time.
Computes all key projectile parameters — range, maximum height, time of flight, impact speed, and impact angle.
Handles multiple unit systems for speed, height, and output, eliminating conversion headaches.
Supports gravity presets for different planets plus custom gravity input for complete flexibility.
Updates results instantly as you adjust inputs, perfect for exploring “what if” scenarios.
Runs entirely client-side with no server communication, keeping your data private and secure.
Free to use on any device with responsive design that works on phones, tablets, and desktops.
Range is the total horizontal distance the projectile travels from launch point to landing point. Trajectory is the entire curved path — the shape of the flight from launch to landing. Range is a single number; trajectory is the complete spatial description of the motion.
Yes, in real-world conditions air resistance significantly affects projectile trajectories. It reduces range, lowers maximum height, and changes the shape from a perfect parabola to an asymmetric curve. This calculator assumes ideal conditions with no air resistance, which is the standard for basic physics calculations.
Without air resistance, the only force acting on a projectile is gravity, which acts vertically. The horizontal motion is uniform (constant speed), while vertical motion is uniformly accelerated. The combination of linear horizontal motion and quadratic vertical motion produces a parabolic path — a mathematical fact proven by the parametric equations.
A 90° launch angle means firing straight up. The projectile goes vertically upward, stops at the peak, then falls straight back down. The range is zero, and the maximum height is h₀ + v₀²/(2g). The trajectory chart shows only a vertical line rather than a curved arc.
Gravity determines the curvature of the trajectory. Higher gravity (Jupiter) creates a tighter, steeper parabola — the projectile falls faster and the arc is more pronounced. Lower gravity (Moon) creates a wider, flatter trajectory — the projectile stays airborne longer and travels further. On the Moon, a 45° launch would go much further than on Earth with the same initial speed.
When launching from an elevated position, the optimal angle for maximum range is less than 45°. The exact optimal angle depends on the ratio of launch height to speed. The calculator can help you find this value by comparing trajectories across different angles.
Yes, this calculator is excellent for analyzing shots in basketball, baseball, golf, and other sports. You can model the trajectory of a ball with a given launch speed and angle, and see whether it clears obstacles or reaches a target. Just remember that air resistance and spin effects are not included.
The calculations are mathematically exact for the assumed conditions — constant gravity, no air resistance, and uniform motion. In ideal conditions, the results are precise. In real-world scenarios, the results are approximations that provide valuable insights but should be verified with field measurements or more complex models.
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