Home / Topics / Kinematics
Trajectory equation
Intermediate
Eliminating time gives y as a function of x.
projectileEliminating t from x = u cosθ·t and y = u sinθ·t − ½gt² gives y = x tanθ − gx²/(2u²cos²θ), a downward parabola. It lets you check if a projectile clears an obstacle at a given x.
At x = 20 m, u = 20 m/s, θ = 45°: y = 20 − 9.8·400/(2·400·0.5) ≈ 10.2 m.
Use the equation only for the path, not for timing.
In more detail
The trajectory equation y = x tan θ − g x² / (2 u² cos² θ) gives the height y at any horizontal position x, and it confirms the parabolic path.
Real-world uses
- Plotting the path of a thrown ball or a kicked football.
- Calibrating water cannons and fountains to hit target points.
- A physicist uses the trajectory equation to fit measured points of a thrown ball and find its launch speed.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
Why is it a parabola?
Because the x-term is linear and the y-term is quadratic in x, a standard parabola form.
Does mass appear in the equation?
No, when air resistance is neglected.
What is the use of the trajectory equation?
It gives the shape of the path directly, so you can read off where the body is at any horizontal distance without finding time.
Related topics in Kinematics
Maximum range at 45° · Horizontal projectile · Projectile on an incline · Velocity at any point of projectile · Effect of air resistance · Relative velocity in 2D