Free Trajectory Calculator

Trajectory of a projectile — often referred to as its flight path — is the curved route an object travels when launched under gravity alone. A Projectile Trajectory Calculator allows you to determine that path quickly by entering just three values: the initial speed, the launch angle, and the starting height above ground. Whether you are studying a baseball arcing through the air, a bullet fired from a weapon, or water streaming from a fountain, this Physics Trajectory Calculator gives both the mathematical formula and a visual plot of the motion.

For most practical situations where air resistance is negligible, the trajectory of a projectile follows a perfect parabola. Everyday examples include a hockey puck in flight, a golf ball after being struck, a jet of water from a fountain, blood droplets in forensic analysis, or an object thrown from a table, plane, building, or bridge. The parabolic shape arises because the only force acting after launch is gravity, which produces a constant downward acceleration.

To derive the trajectory formula, start with the equations of motion for the horizontal and vertical components. The initial velocity $v_{0}$ splits into:

$$v_{0x} = v_{0}\cos(\alpha),\qquad v_{0y} = v_{0}\sin(\alpha),$$

where $\alpha$ is the launch angle measured from the horizontal. Ignoring drag, the horizontal velocity remains unchanged, while the vertical velocity changes linearly with time. The position after time $t$ is:

$$x = v_{0}\cos(\alpha)\,t,\qquad y = v_{0}\sin(\alpha)\,t - \frac{1}{2}gt^{2}.$$

Eliminating $t$ by substituting $t = x/(v_{0}\cos(\alpha))$ into the vertical equation gives the classic Flight Path Calculator result:

$$y(x) = x\tan(\alpha) - \frac{g\,x^{2}}{2v_{0}^{2}\cos^{2}(\alpha)},$$

where $g$ is the gravitational acceleration (approximately $9.81\ \mathrm{m/s^{2}}$ or $32.17\ \mathrm{ft/s^{2}}$). This equation expresses the vertical height $y$ as a function of horizontal distance $x$, confirming that the path is a quadratic parabola.

Using an online Projectile Motion Path Calculator is straightforward. As an example, consider water from a fountain nozzle with a speed of $5\ \mathrm{ft/s}$, a launch angle of $60^{\circ}$, and a nozzle height of $5\ \mathrm{in}$. Simply enter those three values; the tool returns the trajectory equation and a graph showing the full flight path from the nozzle until it hits the ground. One important note: the graph axes may be scaled differently, so the visual angle might appear different from the true launch angle. All calculations assume zero air resistance, which is a reasonable approximation for short‑range, low‑speed projectiles.

This projectile trajectory calculator is ideal for physics homework, engineering design, sports analysis, or any scenario where you need to quickly visualize how changes in velocity, angle, or initial height affect the range and shape of the flight path.