Trajectory Calculator
Find a projectile's position at a given time.
Track a projectile's horizontal and vertical position at any moment in its flight.
How the Math Works
The Trajectory Calculator uses the fundamental equations of projectile motion to determine where an object will be at any given time. The horizontal position x = v₀cosθ·t calculates distance traveled by multiplying the initial velocity's horizontal component (v₀cosθ) by time. The vertical position y = v₀sinθ·t − ½gt² accounts for both upward motion from the initial vertical velocity (v₀sin·t) and downward acceleration due to gravity (½gt²), where g equals 9.8 m/s².
Practical Applications
To use this calculator, input the launch velocity (v₀), launch angle (θ), and time (t) of interest. For example, if you throw a ball at 20 m/s at a 30° angle, the calculator determines its position at 1 second: x = 20cos(30°)·1 ≈ 17.3 m and y = 20sin(30°)·1 − ½(9.8)(1)² ≈ 0.1 m. This works for any projectile motion scenario where air resistance is negligible.
Day-to-Day Use
Understanding projectile trajectories helps in everyday activities like determining if a thrown object will reach a target, planning basketball shots, or calculating safe distances for outdoor activities. Sports coaches use these principles to optimize throwing and kicking techniques, while engineers apply them to design safe ranges for fireworks or construction equipment. Even in video games and sports analysis, trajectory calculations help predict ball paths and improve performance.
Worked example
25 m/s at 40°, t = 1.5 s → x ≈ 28.7 m, y ≈ 13.1 m.
FAQ
When does y become negative?
After the projectile passes its launch height and descends below it.