Trajectory denotes the path that a moving object follows through space and time. In physics and engineering the trajectory is the set of positions an object occupies as a function of time. Everyday examples include a thrown stone, the arc of a soccer ball, or the flight of a shell when a cannon is fired: the precise path depends on initial speed, direction, mass, propellant and forces such as gravity, aerodynamic drag and lift.

Mathematical representation

Trajectories are usually modelled parametrically by a vector function r(t). Velocity v(t) is the first time derivative of r(t) and is tangent to the path; acceleration a(t) is the second derivative and relates to how the direction and speed change. When the forces acting on a body are known, ordinary differential equations determine r(t). In practice numerical integration and state estimation methods (for example Kalman filters) are used to predict or reconstruct trajectories from noisy measurements.

Types and characteristic shapes

  • Projectile — often approximated by a parabola under uniform gravity and negligible air resistance.
  • Ballistic — unguided motion dominated by initial impulse and external forces; common in artillery and meteorites.
  • Orbital — motion under a central inverse-square force produces conic sections (ellipses, parabolas, hyperbolas) in ideal two-body problems.
  • Guided or controlled — trajectories altered in flight by control systems, used in missiles, rockets and spacecraft for transfer, rendezvous and reentry.
  • Stochastic and microscopic — random-walk-like paths in fluid flows or tracks of subatomic particles reconstructed from detector data.

Measurement and reconstruction

Trajectories are measured and reconstructed using sensors such as radar, lidar, high-speed cameras, GPS and motion-capture systems. Data processing fits observed positions to dynamic models, estimates unobserved states, and predicts future motion. Techniques include least-squares fitting, filtering, trajectory smoothing and numerical integration of the governing equations. Accurate reconstruction is essential in accident analysis, ballistics, sports biomechanics and space mission planning.

Applications, history and caveats

Study of trajectories has roots in early experiments on falling bodies and the development of classical mechanics. Today it underpins ballistics, astrodynamics, robotics, animation, navigation and many measurement sciences. Important caveats: trajectory description is frame-dependent (different observers may record different paths), real trajectories deviate from ideal models due to drag, lift, wind, changing mass and control inputs, and in relativistic contexts the natural path of a free particle is described by a geodesic of spacetime rather than a Newtonian trajectory.