Trajectory Calculator
Projectile motion
Results
- Horizontal velocity (Vx)
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- Vertical velocity (Vy)
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- Maximum height
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- Horizontal range (distance)
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Flight path
Enter velocity, angle, and height, then click Calculate to plot the parabolic trajectory.
Hover or drag along the curve to read distance and height at any point.
Figuring out where a thrown object lands should be straightforward, but juggling launch speed, angle, height, and unit conversions on scratch paper gets old fast. A baseball leaving the bat, water arcing from a fountain, or a lab marble rolling off a table all trace the same curved path under gravity—picture the arc of a garden hose set to a medium spray. Enter launch speed, angle, and starting height here to see the parabolic trajectory, horizontal range, peak altitude, and velocity components without manual trig.
How to Use This Trajectory Calculator
- Enter initial velocity (V₀). Launch speed at the instant the projectile leaves the surface—m/s, ft/s, km/h, or mph from the dropdown.
- Enter launch angle (α). Degrees above the horizontal. On flat ground with no air drag, 45° gives maximum range for a given speed.
- Set initial height (h₀). Use 0 from ground level; enter platform, cliff, or table height when the launch starts above the landing surface.
- Click Calculate. Results include Vx, Vy, max height, range, the trajectory equation, and an interactive flight-path chart—hover or drag along the curve to read distance and height at any point.
Trajectory Formulas and Practical Applications
Horizontal motion stays uniform while vertical motion accelerates downward at g ≈ 9.81 m/s². Split the launch speed into components first, then combine motion along each axis.
Vx = V₀·cos(α)
Vy = V₀·sin(α)
Height as a function of horizontal distance:
y = h₀ + x·tan(α) − g·x² / (2·V₀²·cos²(α))
Peak altitude adds the vertical climb to the launch height: hmax = h₀ + Vy² / (2g). On level ground (h₀ = 0), range is R = V₀²·sin(2α) / g.
Worked example: V₀ = 25 m/s, α = 40°, h₀ = 0 gives roughly Vx ≈ 19.2 m/s, Vy ≈ 16.1 m/s, peak near 13.2 m, and range near 60 m. In our testing, plotting the curve beats guessing whether a demo marble clears the far edge of the bench.
Where projectile math shows up
- Sports coaching—estimate carry distance from exit velocity and launch angle.
- Intro physics labs—marble off a table, verify range against measured values.
- Engineering sketches—quick check that a thrown or launched part clears an obstacle.
Standard Units and Conversion Tables
Velocity units
| Unit | Symbol | Conversion to m/s |
|---|---|---|
| Meters per second | m/s | SI default |
| Feet per second | ft/s | 1 ft/s = 0.3048 m/s |
| Kilometers per hour | km/h | 1 km/h ≈ 0.278 m/s |
| Miles per hour | mph | 1 mph ≈ 0.447 m/s |
Height units
Initial height accepts m, cm, ft, and in. Mix freely with velocity units—the tool converts to SI before solving, then displays results in the units you chose.
Frequently Asked Questions
What is the trajectory formula?
y = h₀ + x·tan(α) − g·x² / (2·V₀²·cos²(α))—height as a function of horizontal distance for a projectile launched at speed V₀ and angle α from height h₀.
What shape is a projectile trajectory?
A parabola. Gravity produces constant downward acceleration, so vertical position depends quadratically on time and thus on horizontal distance.
What angle gives maximum range on level ground?
With h₀ = 0 and no air drag, 45° maximizes range for a given launch speed. Launch from a cliff adds height and can push the best angle below 45°.
Does this calculator include air resistance?
No. Real balls and rockets slow in flight; this tool matches textbook problems with constant gravity and no drag. For force and acceleration on a single body, try the Newton's Second Law Calculator or the Force Calculator. Browse all physics tools on RapidRatio.