Trajectory Calculator

The Trajectory Calculator computes the vertical position (height) of a projectile at any given horizontal distance from the launch point. This provides a complete description of the parabolic path, allowing you to determine whether a projectile clears an obstacle, the position of a ball at any point along its flight, or the shape of any ballistic trajectory.

The trajectory equation expresses height $$y$$ as a function of horizontal position $$x$$: $$y = x \tan(\theta) - \frac{gx^2}{2v_0^2 \cos^2(\theta)}$$ This is a parabola opening downward, with the first term representing the linear rise and the second term representing the gravitational pull. The equation is derived by eliminating time from the horizontal and vertical motion equations.

This parabolic form was first identified by Galileo in the 17th century and represents one of the earliest applications of mathematical analysis to physical motion. The equation shows that the trajectory depends on only three parameters: initial speed, launch angle, and gravitational acceleration. All projectiles with the same $$v_0$$ and $$\theta$$ follow identical paths regardless of mass—a profound consequence of the equivalence of gravitational and inertial mass.

The trajectory equation is particularly useful for clearance problems: will a ball clear a wall at distance $$x$$ with height $$h$$? Simply compute $$y(x)$$ and compare with $$h$$. If $$y(x) > h$$, the projectile clears the obstacle. This analysis is essential in sports (goal-kicks over walls, basketball arc over defenders), engineering (fountain design, water jet cutting trajectories), and military applications.

The calculator also determines the velocity vector at the specified position. The horizontal component remains $$v_0 \cos\theta$$ throughout the flight, while the vertical component decreases during ascent and increases during descent: $$v_y = v_0 \sin\theta - gt_x$$, where $$t_x = x/(v_0 \cos\theta)$$ is the time to reach position $$x$$.

When the computed height $$y$$ becomes negative, it indicates the projectile has already hit the ground before reaching that horizontal position. The range where $$y$$ returns to zero gives the maximum horizontal distance. Beyond this point, the trajectory equation gives physically meaningless negative heights that would require the ground to be lower than the launch level.