Projectile Motion Calculator
Analyze an ideal projectile from launch to impact with live range, flight time, apex, velocity components, trajectory data, and a validated Excel export.
Launch inputs
Advanced assumption
Live results
Projectile trajectory
Height is plotted against horizontal distance from launch to impact using the same current-state model as the result cards and workbook.
Flight checkpoints
Five evenly spaced checkpoints make the position and speed changes easy to inspect without relying on the chart alone.
| Time | Horizontal distance | Height | Horizontal velocity | Vertical velocity | Speed |
|---|
How to use the Projectile Motion Calculator
What this calculator does
This calculator estimates ideal two-dimensional projectile motion from an initial speed, launch angle, launch height, gravitational acceleration, and a selected inspection time. It resolves the launch speed into horizontal and vertical components, solves the time at which the projectile reaches the impact level, calculates the horizontal range and maximum height, and reports position and velocity at the selected time. The trajectory chart and checkpoint table show the same motion over the full flight. The model is intentionally idealized: it treats gravity as constant and vertical, and it neglects air resistance, lift, spin, wind, Earth curvature, and terrain shape. For the standard derivation and the separation of horizontal and vertical motion, see the OpenStax treatment of projectile motion.
When to use it
Use the calculator when checking a classroom kinematics problem, comparing how launch angle changes range and apex, estimating the ideal path of a thrown object from an elevated platform, or testing how a different gravitational acceleration would change flight. It is also useful for building intuition: the horizontal component stays constant in this no-drag model while the vertical component decreases at a constant rate.
How to calculate
- The calculator opens with a complete demonstration: 30 m/s initial velocity, 40° launch angle, 2 m launch height, 9.80665 m/s² gravity, and a 2 s inspection time. Results and a validated example workbook are available immediately.
- Choose Metric or US customary. Switching systems converts the current velocity, height, and gravity inputs rather than merely relabeling them.
- Replace Initial velocity (V), Angle of launch (α), and Initial height (h) with your launch conditions. Then set Time for flight parameters to a moment between launch and impact.
- Open Advanced assumption only if you want to change Gravity (g). Read the live results, trajectory, and checkpoint table; all update from one canonical model.
- Select Download Excel to export the current inputs, results, checkpoints, and notes as a real XLSX workbook. Reset clears the demonstration and user-entered numeric data. After Reset, export stays disabled until a complete valid state is entered again.
Input guide
Unit system is required and accepts Metric or US customary. Metric uses m/s, m, and m/s²; US customary uses ft/s, ft, and ft/s². A common mistake is manually converting a value before switching units; the switch already converts the current numbers. Initial velocity (V) is a required nonnegative decimal speed, for example 30 m/s. A zero value represents a drop or free fall when the initial height is above the impact level. Increasing it generally increases range, flight time, and maximum height, with the exact effect depending on angle. Enter ordinary decimal notation with a dot; grouped numbers, decimal commas, and scientific notation are rejected to avoid ambiguity.
Angle of launch (α) is required and must be from 0° through 90°, for example 40°. A 0° launch begins horizontally, while 90° is vertical and therefore has essentially zero horizontal range. Higher angles increase the vertical component but reduce the horizontal component; the range effect is therefore not monotonic. Initial height (h) is required and may be zero or positive, for example 2 m. A higher launch point normally increases flight time and range because the projectile has farther to fall before reaching the impact level.
Time for flight parameters is a required nonnegative time in seconds, for example 2 s. It must not exceed the computed time of flight. It does not change the trajectory itself; it only chooses the instant used for the position and velocity readouts. Gravity (g) is a required positive acceleration. The startup value 9.80665 m/s² is the conventional standard acceleration of free fall listed by NIST's Guide to the SI. Lower gravity lengthens flight and usually increases range and apex; higher gravity does the opposite. Local gravity can differ slightly, so use a different value only when that distinction matters.
Output guide
Distance (d) is the horizontal range from launch to impact, shown in metres or feet. Time of flight (t) is the total airborne time in seconds. Maximum height (hmax) is the highest elevation above the impact level, including the initial height. Initial horizontal velocity (Vx) and Initial vertical velocity (Vy) are the cosine and sine components of the launch speed. The Horizontal position at selected time, Height at selected time, Vertical velocity at selected time, and Speed at selected time describe the projectile at the time you entered. Vertical velocity can be negative after the apex; a negative value means downward motion, not a negative speed.
The three summary pills repeat range, flight time, and apex so the principal outputs remain visible at a glance. The Projectile trajectory line series plots height against horizontal distance over the entire current flight; it is omitted for a vertical launch because multiple x positions do not exist. The Flight checkpoints table contains Time, Horizontal distance, Height, Horizontal velocity, Vertical velocity, and Speed at 0%, 25%, 50%, 75%, and 100% of total flight time. These values are estimates from the ideal equations, not measurements of a real object.
Worked example
With the startup inputs, the launch components are Vx = 30 cos(40°) = 22.981 m/s and Vy = 30 sin(40°) = 19.284 m/s. Solving 0 = 2 + 19.284t – ½(9.80665)t² gives a time of flight of about 4.034 s. Multiplying that time by the constant horizontal component gives a range of 92.704 m. The maximum height is 2 + Vy²/(2g), or 20.959 m. At the selected 2 s inspection time, the model gives x = 45.963 m, y = 20.954 m, vertical velocity – 0.330 m/s, and total speed 22.984 m/s. These are the same startup values written into the downloadable workbook.
Learn more
A useful conceptual check is that horizontal and vertical motions are independent in this ideal model: gravity changes only the vertical component, while the horizontal component remains constant. The Physics Classroom explanation of horizontal and vertical velocity components develops that idea with diagrams and examples.
Model, formulas, and interpretation
The calculator converts the active unit system to SI internally, performs the physics in canonical units, then converts displayed lengths and speeds back to the active system. For launch speed V and angle α, the initial components are Vx = V cos α and Vy = V sin α. Horizontal position is x(t) = Vx t, while height is y(t) = h + Vy t – gt²/2. The positive root of y(t) = 0 gives impact time. Range is then Vx × tflight.
The apex calculation deserves one nuance. When the launch angle is between 0° and 90°, the vertical velocity reaches zero after launch, so the added rise is Vy²/(2g). With a horizontal launch, that rise is zero and the initial height is already the maximum height. A purely vertical launch remains a valid projectile case, but horizontal range is zero; the calculator therefore reports the scalar results and suppresses the line chart because an x-versus-y trajectory comparison would collapse to one horizontal position.
Limits and common mistakes
- Air resistance can materially shorten range and change the trajectory, especially for light, slow, or high-drag objects. The ideal equations do not model that effect.
- The impact level is fixed at zero height. If your target sits above or below that level, represent the launch height relative to the target elevation or use a model with an explicit target-height parameter.
- Do not interpret maximum range rules mechanically when the launch height is nonzero. The familiar 45° result applies to equal launch and landing elevations under the no-drag model.
- Use consistent units. The unit switch handles conversion for you; typing a value in feet while Metric is selected will be treated as metres.
Projectile calculations are useful because they show how a single two-dimensional motion can be decomposed into simpler one-dimensional motions. The practical value is not just the final range number: checking the velocity components, apex, selected-time state, and flight checkpoints together makes it easier to detect an inconsistent assumption or an input-unit mistake.