Projectile Motion Calculator
Calculate peak height, flight time, horizontal range, and impact velocity for projectile motion and horizontal projection from initial speed, angle, initial height, and gravity. Includes SVG trajectory plots, worked solutions, and state at time t.
Examples & Presets
−89° to 90° (0°: horizontal, 90°: vertical upward, negative: downward)
0 m for ground level. Enter height if launched from a cliff or building
Results
Horizontal range R
—
Total horizontal distance
Flight time
—
Total duration until ground impact
Max height y_max
—
—
Time to peak
—
Instant vy = 0
Impact speed
—
—
Impact angle
—
Angle with ground
Horizontal initial v₀x
—
v₀·cos θ (constant)
Vertical initial v₀y
—
v₀·sin θ
Trajectory plot (SVG)
Scale unit: mState at time t simulation
Drag the slider to inspect the position and velocity at any point during flight.
x(t)
—
y(t)
—
v(t)
—
Angle
—
Worked solutions & Derivations
Note: This calculator assumes ideal projectile motion without air resistance, Coriolis force, or Earth curvature. Impact velocity magnitude is verified against conservation of mechanical energy.
Done
Projectile Motion Calculator
How to use
- 1
Enter conditions
Set initial speed (m/s or km/h), angle (−89° to 90°), starting height (m), and gravity, or pick an example preset.
- 2
Check results & trajectory
Instantly view horizontal range, flight time, peak height, impact speed, and the dynamic SVG plot.
- 3
Inspect solutions & simulate
Review worked equations, energy conservation verification, and drag the slider to track position and velocity at time t.
Features
- Instantly solve projectile motion from launch speed, angle, initial height, and gravity
- Handles ground launches (h₀=0), elevated launches (h₀>0), horizontal throws (0°), vertical throws (90°), and downward launches
- Interactive SVG trajectory plot featuring launch, apex, impact, and dynamic current-position markers
- Interactive time slider to inspect position (x, y) and velocity (v, θ) at any instant t
- Worked step-by-step kinematic solutions cross-checked with conservation of mechanical energy
- One-click copy of conditions and results; runs 100% locally in your browser
Use cases
Physics homework and problem sets
Verify kinematic homework problems step by step and confirm formula substitutions.
Lab reports & experiments
Compute theoretical values for projectile range, flight duration, and trajectory curves.
Game development ballistics
Prototype launch velocities, angles, and trajectories directly in the browser.
Details
Projectile motion describes an object moving in two dimensions under gravity alone. Horizontally with no net force, it travels at constant velocity (vx = v₀·cos θ). Vertically under downward acceleration g, it undergoes constant acceleration (vy = v₀·sin θ − gt). By the independence of perpendicular motions (Galileo's principle), these components can be analyzed separately.
At the apex (peak), the vertical velocity momentarily vanishes (vy = 0). The time to peak is t_peak = (v₀·sin θ) / g, and the maximum height is y_max = h₀ + (v₀·sin θ)² / (2g). Note that the horizontal velocity vx remains constant, so overall velocity at the top is not zero.
Total flight time until ground impact (y = 0) is found by solving the quadratic equation y(t) = h₀ + (v₀·sin θ)t − ½gt² = 0. When launched from the ground (h₀ = 0), flight time is t_flight = 2(v₀·sin θ) / g, exactly twice the time to peak. Horizontal range is R = vx × t_flight.
For launches from ground level (h₀ = 0), range simplifies to R = (v₀²·sin 2θ) / g, achieving maximum distance at θ = 45° where sin 2θ = 1. However, when starting from an elevated position (h₀ > 0), the optimal launch angle for maximum range is always less than 45°.
Impact speed satisfies conservation of mechanical energy: ½m(v₀)² + mgh₀ = ½m(v_land)² yielding v_land = √(v₀² + 2gh₀), perfectly matching the kinematic equations. Both methods are presented side by side for verification.
FAQ
Why is 45 degrees the optimal angle for maximum projectile range?
For ground launches (h₀ = 0), the horizontal range formula is R = (v₀²·sin 2θ) / g. Since the sine function reaches its maximum value of 1 at 90°, setting 2θ = 90° yields θ = 45°.
Is 45 degrees still optimal when launched from an elevated cliff or building?
No, when h₀ > 0, the optimal launch angle is always less than 45° (typically between 30° and 40° depending on height). Because the initial height naturally provides extra flight time, investing more velocity into the horizontal component results in greater total distance.
Is the projectile's velocity zero at the peak?
No, only the vertical component of velocity is zero (vy = 0). The horizontal velocity component (vx = v₀·cos θ) remains constant throughout the flight. Thus, at the peak, the projectile moves horizontally at speed vx.
If a ball is dropped and another is fired horizontally from the same height, which hits the ground first?
Ignoring air resistance, they hit the ground at the exact same instant. Because perpendicular motions are independent, the horizontal speed does not alter the vertical equation of motion y(t) = h₀ − ½gt².
Does this calculator account for air drag?
No, this tool models ideal Newtonian projectile motion without air resistance, as taught in introductory physics. In real air, drag forces shorten the range and cause the trajectory to become asymmetric.
Can I change the gravitational acceleration g?
Yes. You can select standard textbook Earth gravity (9.8 m/s²), ISO standard gravity (9.80665 m/s²), the Moon (1.62 m/s²), Mars (3.71 m/s²), or enter any custom gravity value.
Related guide
GuideProjectile Motion & Horizontal Launch: Formulas, Steps & Examples
Solve projectile motion and horizontal launches by separating horizontal constant-velocity motion from vertical constant-acceleration motion. Covers time of flight, peak height, range, elevated launches from cliffs, why 45 degrees maximizes range, and energy-conservation verification with worked examples.
GuideHow to Solve Quadratic Equations: The Formula and Discriminant
Learn to solve any quadratic equation by substituting into the quadratic formula, read the discriminant to predict the number of roots, and know when factoring is faster. Worked examples with full substitutions, common mistakes and a free calculator.
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Verified: Cross-checked against independent calculations (45° ground launch giving R=39.2m and t=2.828s, horizontal throw from 19.6m giving R=40m and t=2.0s, elevated 30° throw from 39.2m giving R=44.03m with energy conservation verification, vertical 90° throw, km/h unit conversion) plus error handling are covered by browser tests
Did you know?
An object launched horizontally and one dropped straight down from the same height hit the ground at the exact same instant. As Galileo demonstrated in his 1638 Two New Sciences, perpendicular motions are independent: horizontal velocity does not change the vertical time of fall.