Trajectory Calculator – Projectile Motion Range and Height
Calculate horizontal range, maximum height, and time of flight for any projectile from initial velocity, launch angle, and starting height.
Enter initial velocity, launch angle (0–90°), and initial height. Choose metric (m, m/s) or imperial (ft, ft/s) units for instant trajectory analysis.
Trajectory Calculator – Projectile Motion Range and Height
Calculate horizontal range, maximum height, and time of flight for any projectile from initial velocity, launch angle, and starting height.
About the trajectory calculator
Projectile motion is one of the most studied problems in classical mechanics. When an object is launched into the air under the influence of gravity alone — with air resistance neglected — its path traces a smooth parabolic arc called a trajectory. This calculator uses the standard kinematic equations of projectile motion to compute the three key outputs engineers, athletes, and physicists need most: maximum height, horizontal range, and total time of flight.
The motion is decomposed into two independent components. In the horizontal direction there is no acceleration (ignoring drag), so the object travels at the constant horizontal velocity v₀ₓ = v₀·cos α throughout its flight. In the vertical direction the object experiences a constant downward acceleration g — equal to 9.81 m/s² near Earth's surface in metric units, or 32.2 ft/s² in imperial units. The vertical velocity at any instant is v_y = v₀y − g·t, where v₀y = v₀·sin α.
When the object is launched from a height h above the landing surface, the time of flight is found by solving the quadratic: 0 = h + v₀y·t − ½g·t². The positive root gives t = (v₀y + √(v₀y² + 2gh))/g. The horizontal range follows immediately as R = v₀ₓ·t. The maximum height is reached when the vertical velocity equals zero, occurring at t_peak = v₀y/g; substituting back gives H_max = h + v₀y²/(2g).
A widely cited rule of thumb says the optimal launch angle for maximum range is 45°. This is only correct when the launch and landing heights are equal. When a projectile is launched from an elevation — say a cannon on a hill — the optimal angle is less than 45°. Conversely, when launched upward toward a higher landing point, the optimal angle exceeds 45°. This calculator handles all three scenarios through its initial height input.
Practical applications are broad: sports science uses trajectory analysis to optimise ball kicks, throws, and shots; ballistics engineers apply the same equations to artillery, missiles, and small arms; video game and simulation developers use projectile physics for realistic object movement; and safety engineers calculate the throw distance of fragments in explosion scenarios. The metric/imperial toggle makes the calculator equally useful in research contexts and in countries that use the US customary system.
Trajectory calculator examples
Three scenarios demonstrating metric and imperial units across different launch conditions.
| Input | Range | Notes |
|---|---|---|
| v₀=100 m/s, α=30°, h=0 m (metric) | Range ≈ 882.9 m, H_max ≈ 127.4 m | Classic cannonball scenario. At 30° the range is 882.9 m and max height 127.4 m; time of flight is 10.19 s. |
| v₀=70 m/s, α=15°, h=0.05 m (metric) | Range ≈ 249.9 m, H_max ≈ 16.8 m | Golf drive. Drivers typically launch at 9–15°; low angle trades height for distance on a flat fairway. |
| v₀=90 ft/s, α=45°, h=6 ft (imperial) | Range ≈ 257.4 ft, H_max ≈ 68.9 ft | Baseball throw from 6 ft above ground. Imperial units show the range and height in feet for direct field comparison. |
How to use the trajectory calculator
- Select your preferred unit system — Metric (metres, m/s) or Imperial (feet, ft/s). Gravity is automatically set to 9.81 m/s² or 32.2 ft/s².
- Enter the initial velocity (the speed at which the object leaves the launch point) as a positive number.
- Enter the launch angle in degrees between 0° and 90°. A 0° angle means purely horizontal launch, 90° means straight up.
- Enter the initial height — the vertical distance above the ground level where the object will land. Use 0 for a flat surface and a positive number for an elevated launch point.
- Click Calculate Trajectory. The calculator returns horizontal range, maximum height, time of flight, and horizontal and vertical velocity components.
Trajectory calculator FAQ
Why is 45° not always the optimal launch angle?
The 45° rule applies only when the launch and landing heights are identical. If you launch from an elevation above the landing point, the optimal angle is less than 45°. If you are launching upward toward a higher landing point, the optimal angle is greater than 45°. The exact optimum can be derived by differentiating the range formula with respect to angle and setting the result to zero.
Does air resistance affect the results?
This calculator uses ideal projectile motion equations with no air drag. In reality, air resistance reduces range and maximum height — sometimes significantly for light or fast-moving projectiles like golf balls, bullets, or shuttlecocks. For engineering work requiring drag modelling, you would need numerical integration with a drag coefficient term.
What is the difference between time of flight and time to peak height?
Time to peak height is t_peak = v₀y/g, the moment when vertical velocity reaches zero and the object is momentarily stationary in the vertical direction. Time of flight is the total time until the projectile lands, which equals t_peak plus the descent time back to the landing altitude. When the initial height equals zero, the descent takes exactly as long as the ascent.
How can I convert the result to kilometres or miles?
The metric result is in metres; divide by 1000 for kilometres. The imperial result is in feet; divide by 5280 for miles, or divide by 3.281 to convert feet to metres. The velocity components are in m/s (metric) or ft/s (imperial); multiply m/s by 3.6 for km/h or by 2.237 for mph.
Can I use this for objects thrown horizontally?
Yes — set the launch angle to 0°. With a horizontal launch the initial vertical velocity is zero, so the time of flight is entirely determined by the initial height: t = √(2h/g). The horizontal range is then simply v₀ × t. This is the classic scenario for objects rolling off a table or jumping from a cliff.
What gravitational constant does the calculator use?
For metric calculations, the calculator uses g = 9.81 m/s², the standard gravitational acceleration at sea level. For imperial calculations it uses g = 32.2 ft/s². Both values are accurate for most Earth-surface applications. Calculations on other planets or at high altitude would require a different g value.