Projectile Motion Calculator for Range, Height and Flight Time
Model ideal projectile motion from launch speed, angle, height and gravity. Calculate a complete flight, inspect position and velocity at a chosen time, solve low and high target angles, or find the speed required at a fixed angle.
- Four calculation modes
- Unequal launch and landing heights
- Target-angle solver
- Responsive trajectory plot
Online Projectile Motion Calculator
Choose a method, enter compatible values and review the trajectory, velocity components, calculation steps and assumptions. The engine calculates in SI units before converting the display.
The trajectory and component results will appear here.
Trajectory Results
| Quantity | Symbol | Value |
|---|---|---|
| Result | — | No current result |
Trajectory Plot
The horizontal and vertical axes are scaled independently so the path remains readable.
Calculation Steps
- Select a method and enter its active values.
- The calculator will resolve the motion into horizontal and vertical components.
How to Use This Projectile Motion Calculator
- Choose the calculation method. Select a complete flight, a state at time, target angles or required launch speed.
- Set the coordinate meaning. Angles are measured from forward horizontal, positive height points upward and gravity is entered as a positive magnitude.
- Enter the known motion values. Use launch speed, angle, heights, elapsed time or target coordinates for the selected method.
- Choose each input unit. The calculator converts speed, distance, time and gravity to SI units before solving.
- Select output units and precision. Internal values remain unrounded while the displayed result uses your chosen significant digits.
- Press Calculate Motion. Review the main answer, trajectory table, plot, component equations and physical interpretation.
- Check the model limits. Confirm that constant gravity and negligible drag, wind, spin, lift and terrain are acceptable for your use.
Projectile Motion Formulas and Coordinate Convention
Projectile motion separates into two independent components when gravity is constant and air resistance is ignored. The calculator defines forward as positive x and upward as positive y. The entered gravitational acceleration g is a positive magnitude, so vertical acceleration is −g.
The horizontal component stays constant in this model. The vertical component changes uniformly. Total speed is the magnitude of the velocity vector, found with v = √(vx2 + vy2). The direction is atan2(vy, vx), which preserves the correct upward or downward sign. Direction is undefined when both velocity components are zero.
Launch height and landing height must use one shared reference level. For example, if ground is zero and a platform is 12 m high, enter 12 m for the launch height and 0 m for the landing height. You may choose another datum, but both heights must change together. Target modes instead use a relative height: zero means level with launch, a positive value means above launch and a negative value means below launch. This distinction prevents a common error in unequal-height problems.
What Each Calculation Mode Solves
Complete flight to a landing height
This mode solves the time when the projectile reaches the entered landing height. It supports a landing surface above, equal to or below the launch height. When a higher level crosses the trajectory twice, the calculator shows the earlier ascending crossing and uses the later descending crossing as the flight endpoint. A level that touches the apex has one repeated root and is reported as tangent contact. The mode also reports range, time to apex, maximum height and endpoint velocity.
Position and velocity at a time
Enter an elapsed time to evaluate x, y, horizontal velocity, vertical velocity, total speed and direction. The result is a mathematical state. It does not stop automatically at ground or another obstacle because no collision height is entered in this mode.
Launch angles to reach a target
For a forward horizontal distance, relative target height and fixed speed, there can be two angles, one tangent angle or no solution. The low path is flatter and faster to the target. The high path spends more time in the air. The calculator also gives the theoretical minimum speed and its matching angle.
Required speed at a fixed angle
This inverse mode solves the initial speed required to pass through the target at the chosen angle. It stops when the geometry is impossible, such as a trajectory that does not rise enough to meet the target at that angle.
Target-equation check
Eliminating time between the horizontal and vertical equations gives the no-drag target relationship below, where X is forward distance and Δy is target height relative to launch:
For a fixed speed, this becomes a quadratic in tan θ. A positive discriminant gives low and high paths, zero gives one tangent path, and a negative discriminant means the target is unreachable at that speed. The theoretical minimum speed is vmin = √[g(√(X2 + Δy2) + Δy)]. The calculator evaluates the equivalent stable form when the target is far below launch, avoiding subtraction that could erase a small valid result.
Flight Time, Range and Maximum Height
Let yL be the landing height. Substituting yL into the vertical position equation produces a quadratic in time. The later nonnegative solution normally describes the descending landing-height crossing; a repeated positive root instead means the selected level touches the trajectory at the apex:
The calculator uses a cancellation-resistant equivalent for a downward launch toward a lower landing level. If the expression under the square root is negative, the landing level lies above the attainable apex. No real crossing exists.
The apex formulas apply when the initial vertical component is positive. For a horizontal or downward launch, the highest point after release is the launch point at time zero. Range means horizontal displacement to the selected landing height, not the curved distance travelled along the path.
Worked Projectile Motion Examples
Example 1: 20 m/s at 45 degrees on level ground
With v0 = 20 m/s, θ = 45°, equal launch and landing heights and g = 9.80665 m/s², both initial components equal 14.1421 m/s.
The ideal impact speed is again 20 m/s because the projectile returns to its launch height without dissipative forces.
Example 2: Horizontal launch from 20 metres
A projectile leaves horizontally at 10 m/s from y0 = 20 m and lands at zero. Its initial vertical velocity is zero.
The impact vertical velocity is −19.8033 m/s and the total impact speed is about 22.1871 m/s.
Example 3: Two angles to a level target
At 20 m/s, a target 20 m forward at the launch height has two ideal solutions under standard gravity:
Both paths reach the same point. The low path arrives sooner, while the high path rises farther and remains airborne longer.
Supported Units and Gravity
The engine converts all active inputs to metres, seconds and metres per second before calculating. Exact defined conversion factors are used for the international foot, yard, mile, miles per hour and knot. Output conversion occurs only after the physical result is complete.
| Quantity | Supported units | SI basis |
|---|---|---|
| Distance and height | mm, cm, m, km, ft, yd, mi | metre |
| Speed | m/s, km/h, mph, ft/s, kn | metre per second |
| Time | ms, s, min, h | second |
| Acceleration | m/s², ft/s², g0 | metre per second squared |
| Angle | degree | converted to radians internally |
The default 9.80665 m/s² is exact standard gravity, a defined reference rather than the measured value at every location. Use the value stated in your assignment, experiment or site data. A common classroom approximation of 9.8 m/s² produces slightly different results.
Accuracy, Edge Cases and Model Limits
The calculator accepts finite decimals and scientific notation. It rejects malformed entries, impossible target geometry, unsupported angles, nonpositive gravity and arithmetic outside the supported browser range. A zero initial speed is allowed for a dropped object in forward modes. Inverse target modes require positive speed or positive horizontal distance.
Vertical launches at exactly ±90° are handled without dividing by a horizontal component. A horizontal launch at the same landing height has only the release-time intersection, so it has no positive flight interval. A target at the exact apex produces one tangent solution. Editing an active field clears the old result, table and plot so stale output is not mistaken for a new calculation.
The ideal equations ignore air resistance, lift, wind, spin, changing orientation, buoyancy, terrain, Earth rotation, curvature and variation in gravity with height. These effects matter for many balls, long flights, high speeds and large altitude changes. A real trajectory can be shorter, asymmetric and displaced sideways.
Do not use this page for firearm aiming, weapons, aviation, launch operations, fall protection or other safety-critical decisions. Those uses require validated environmental data, drag and lift models, equipment characteristics, legal compliance and qualified review.
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Projectile Motion Calculator FAQs
What is projectile motion?
Projectile motion is two-dimensional motion under gravity after launch. In the ideal model, horizontal velocity remains constant while vertical velocity changes at a constant downward rate. Air resistance, wind and lift are excluded.
How do you calculate projectile range?
First find the valid flight time from the vertical position equation. Then multiply it by the horizontal velocity component: R = v0 cos(theta) times t. The familiar v0 squared sin(2 theta) divided by g shortcut applies only when launch and landing heights are equal.
How do you calculate time of flight?
Set the vertical position equation equal to the landing height and solve the resulting quadratic for time. Use the later nonnegative root unless the equation has one repeated positive root at the apex. If the landing level is above the apex, no real flight-time solution exists.
How do you find maximum height?
When the initial vertical component is positive, the rise above launch is v0y squared divided by 2g, and the apex time is v0y divided by g. For a horizontal or downward launch, the maximum after release is the launch height.
How are horizontal and vertical velocity components calculated?
The initial components are v0x = v0 cos(theta) and v0y = v0 sin(theta). Without drag, v0x stays constant. At time t, vertical velocity is vy = v0y minus gt.
How do you calculate a horizontal launch from a height?
Set the launch angle to zero and enter a launch height above the landing height. Flight time is the square root of twice the vertical drop divided by g. Horizontal range equals horizontal speed multiplied by that time.
Does a 45-degree angle always give maximum range?
No. A 45-degree angle gives maximum ideal range only when launch and landing heights are equal, gravity is constant and drag is absent. A different height, air resistance or another constraint changes the best angle.
How are impact speed and impact angle calculated?
At impact, horizontal velocity remains v0x and vertical velocity is v0y minus gt. Impact speed is the vector magnitude sqrt(vx squared plus vy squared). The signed impact direction is atan2(vy, vx); if both components are zero, direction is undefined.
What gravity value should I enter?
Use the value stated by your problem or appropriate for the location. The default 9.80665 m/s squared is exact standard gravity. It is a reference value, not the measured local acceleration at every point on Earth.
Does this calculator include air resistance?
No. It uses an ideal vacuum-style model with constant gravity. Real drag, wind, spin and lift can materially change range, height, flight time and impact velocity, especially for light, fast or irregular objects.
Method References
- OpenStax University Physics, Projectile Motion, independent component equations, maximum height and unequal-height examples.
- NASA Glenn, Ballistic Flight Equations, ideal ballistic position and velocity relationships.
- NASA Glenn, Flight Equations with Drag, why real trajectories depart from the no-drag model.
- BIPM Declaration on Standard Acceleration of Gravity, the exact 9.80665 m/s² reference value.
- NIST Guide to the SI, Appendix B.8, length and speed conversion factors.
- BIPM SI Brochure, 9th edition, SI units and coherent quantity notation.
Disclaimer: This calculator provides educational and preliminary results from an ideal constant-gravity, no-drag model. Verify coordinates, units, initial conditions, environmental effects, collision surfaces and an appropriate physical model before relying on any result. Do not use it for weapons, aviation, launch operations or other safety-critical decisions.