Projectile Motion Calculator

Free Online Projectile Motion Calculator with Steps

This projectile motion calculator instantly finds the range, maximum height, time of flight, and impact speed of any object launched into the air at an angle — a football, a cannonball, a basketball shot, an arrow, or a rocket in a physics problem. Enter the initial velocity, the launch angle above the horizontal, an optional launch height, and gravity (Earth, Moon, Mars, or a custom value), and the calculator instantly resolves the motion into horizontal and vertical components, applies the standard kinematics equations, and shows a complete, copyable step-by-step solution alongside a fully labeled trajectory diagram plotted to scale from your exact numbers.

Whether you're a physics student working through projectile motion for the first time, checking homework before an exam, or an engineer or sports analyst estimating how far or how high something will travel, this tool handles launches from ground level as well as launches from an elevated height (a cliff, a building, or a raised platform), and always shows exactly how each result was calculated — not just the final numbers.

What Is Projectile Motion?

Projectile motion describes the curved path traced by any object that is launched into the air and then moves only under the influence of gravity, with air resistance ignored. Once launched, the object's horizontal velocity stays constant the entire flight, while its vertical velocity is continuously slowed, brought to zero at the peak, and then accelerated back downward by gravity — producing the familiar symmetric, parabolic arc you see in the diagram above.

Because the horizontal and vertical motions are completely independent of each other, projectile motion problems are almost always solved by breaking the initial velocity into an x-component and a y-component, solving each direction separately with ordinary constant-acceleration kinematics, and then recombining the results at the end — exactly the approach this calculator uses and shows step by step.

Projectile Motion Formulas

Given an initial speed v0 launched at angle θ above the horizontal from height h0, with gravitational acceleration g, the key formulas are:

• Horizontal velocity component: vx = v0 · cos(θ) — stays constant throughout the flight
• Vertical velocity component: vy = v0 · sin(θ) — changes continuously due to gravity
• Time to reach maximum height: t_peak = vy ÷ g
• Maximum height: H = h0 + vy² ÷ (2g)
• Time of flight (returns to y = 0): T = [vy + √(vy² + 2·g·h0)] ÷ g (simplifies to T = 2vy ÷ g when h0 = 0)
• Horizontal range: R = vx × T
• Position at any time t: x(t) = vx·t and y(t) = h0 + vy·t − ½·g·t²

How to Solve a Projectile Motion Problem Step by Step

Every projectile motion problem, however it's phrased, can be solved with the same sequence of steps:

• Step 1 — List the knowns: initial speed v0, launch angle θ, launch height h0 (0 if not stated), and gravity g (9.81 m/s² on Earth unless stated otherwise).
• Step 2 — Resolve v0 into horizontal and vertical components using vx = v0cos(θ) and vy = v0sin(θ).
• Step 3 — Use the vertical component to find the time to the peak (t_peak = vy/g) and the maximum height (H = h0 + vy²/2g).
• Step 4 — Find the total time of flight T by solving for when the height returns to zero.
• Step 5 — Multiply the constant horizontal velocity by the total flight time to get the range: R = vx × T.
• Step 6 — If needed, find the impact velocity by combining the unchanged horizontal speed with the vertical speed at landing (vy − g·T).

Why the Trajectory Is a Parabola

Substituting t = x/vx into the vertical position equation y(t) = h0 + vy·t − ½g·t² produces y as a function of x that has an x² term, an x term, and a constant — the defining shape of a parabola. This is why every projectile's path, from a thrown ball to a fired artillery shell, traces the same smooth, symmetric curve, always opening downward because gravity is the only force acting on it after launch (ignoring air resistance).

The diagram on this page plots that exact parabola using your real numbers, so you can see visually how the launch angle and speed determine the shape, height, and width of the arc, with the peak and landing point both marked and labeled with their calculated values.

The Optimal Launch Angle for Maximum Range

When the launch and landing heights are equal (h0 = 0), the range formula simplifies to R = v0²·sin(2θ)/g, and since sin(2θ) reaches its maximum value of 1 when 2θ = 90°, the range is maximized at a launch angle of exactly 45°. Any angle above or below 45° — say 30° or 60° — produces the same range as its complementary angle, because sin(2×30°) = sin(2×60°) = sin(60°). This is a classic and frequently tested result in introductory physics.

When the launch height is above the landing height (h0 > 0), however, the optimal angle for maximum range is always slightly less than 45°, since the projectile has extra time to travel horizontally while it falls that additional height. This calculator handles both cases correctly by using the general time-of-flight formula that accounts for a nonzero launch height.

How to Use This Projectile Motion Calculator

Enter the initial launch speed in your preferred unit (meters/second, kilometers/hour, miles/hour, feet/second, or knots), the launch angle in degrees above the horizontal (0° is straight along the ground, 90° is straight up), and the launch height if the object starts above the landing surface (leave it at 0 for ground-level launches). Choose Earth, Moon, or Mars gravity from the dropdown, or select Custom to enter any gravitational acceleration you like — useful for other planets, or for classroom problems that specify g = 10 m/s² for simplicity.

The calculator instantly shows the range, maximum height, total time of flight, time to reach the peak, velocity components, and impact speed and angle, all backed by a complete step-by-step solution and a trajectory diagram plotted to scale with every key value labeled directly on the chart.

Worked Examples

Example 1 — a football kicked at 25 m/s at 45°: vx = 25cos(45°) = 17.68 m/s, vy = 25sin(45°) = 17.68 m/s. Time of flight T = 2×17.68/9.81 = 3.60 s. Range = 17.68 × 3.60 = 63.7 m. Maximum height = 17.68²/(2×9.81) = 15.9 m.

Example 2 — a ball thrown from a 40 m cliff at 15 m/s at 20°: vx = 15cos(20°) = 14.10 m/s, vy = 15sin(20°) = 5.13 m/s. Because h0 = 40 m is not zero, the time of flight uses the full formula: T = [5.13 + √(5.13² + 2×9.81×40)] / 9.81 ≈ 3.36 s, giving a range of about 47.4 m and a maximum height of about 41.3 m above the ground.

Real-World Applications of Projectile Motion

Projectile motion isn't just a textbook exercise — the same equations show up across sports, engineering, and everyday physics:

• Sports science: analyzing the optimal launch angle for a javelin throw, a basketball shot, a golf drive, or a long jump.
• Ballistics and defense: predicting the range and impact point of a fired projectile.
• Firefighting and irrigation: calculating how far a water jet from a hose or sprinkler will travel at a given angle and pressure.
• Space and aerospace engineering: modeling suborbital trajectories and payload drop points, adjusted for different planetary gravities.
• Civil and mechanical engineering: designing safety barriers, catapults, and other systems that launch or must contain moving objects.
• Video game and animation physics engines: simulating realistic arcs for thrown or launched objects.

Tips for Solving Projectile Motion Word Problems Faster

Always split the problem into horizontal and vertical parts immediately — trying to solve for range or height without first finding vx and vy is the most common source of errors. Remember that horizontal velocity never changes during the flight, since gravity has no horizontal component, so vx at launch equals vx at landing. Watch out for problems that give an angle below the horizontal (a downward launch) or ask for quantities partway through the flight rather than at landing — in both cases, use the general position equations x(t) and y(t) directly rather than the shortcut range/height formulas, which only apply to the full flight. And double-check your angle is in the range 0°–90°, since angles outside that range don't correspond to a physical upward launch.