Trajectory Projectile Motion
Plot the parabolic trajectory of any projectile from launch velocity, angle, and height. Free physics calculator with interactive chart showing flight time, range, and max height.
About This Calculator
The Trajectory Projectile Motion Calculator helps physics students, engineers, and enthusiasts analyze the parabolic flight path of any projectile launched at an angle. By entering the initial velocity, launch angle, and optional initial height, you get the complete trajectory equation, horizontal and vertical velocity components, time of flight, maximum range, and peak height — all with an interactive parabolic trajectory chart showing the full flight path.
This calculator uses the standard kinematic equations for projectile motion under constant gravitational acceleration (g = 9.80665 m/s²). The trajectory is described by the equation y = h₀ + x·tan(α) − g·x²/(2·v₀²·cos²(α)), where h₀ is the initial height, α is the launch angle, and v₀ is the initial velocity. Air resistance is neglected, making this ideal for textbook problems and introductory physics courses.
Key Concepts
The horizontal velocity component Vx = v₀·cos(α) remains constant throughout the flight (no horizontal forces). The vertical velocity component Vy = v₀·sin(α) − g·t changes linearly due to gravity. The projectile reaches maximum height when Vy = 0, at time t = Vy₀/g. The range is the total horizontal distance traveled when the projectile returns to its initial height (or hits the ground from initial height h₀). The optimal launch angle for maximum range is 45° when launched from ground level.
Regional Notes
All regions: The calculator uses metric units (m/s, meters, degrees) and the standard gravitational acceleration of 9.80665 m/s². These units are consistent with the International System of Units (SI) used in physics education worldwide. The results are independent of currency or regional formatting since this is a pure physics calculation.
Frequently Asked Questions
What is the trajectory of a projectile?
The trajectory of a projectile is the parabolic path a projectile follows under the action of gravity. It is described by the equation y = h₀ + x·tan(α) − g·x²/(2·v₀²·cos²(α)), where h₀ is initial height, α is launch angle, v₀ is initial velocity, and g is gravitational acceleration (9.80665 m/s²).
How do you calculate projectile trajectory?
Projectile trajectory is calculated using the parametric equations of motion: horizontal position x = v₀·cos(α)·t and vertical position y = h₀ + v₀·sin(α)·t − ½·g·t². By eliminating time t, you get the trajectory equation y = h₀ + x·tan(α) − g·x²/(2·v₀²·cos²(α)).
What is the optimal angle for maximum range?
The optimal launch angle for maximum horizontal range is 45 degrees when launched from ground level (initial height = 0). This is because the range equation R = v₀²·sin(2α)/g reaches its maximum when sin(2α) = 1, which occurs at 2α = 90° or α = 45°.
Does air resistance affect projectile trajectory?
This calculator neglects air resistance. In real-world conditions, air resistance (drag) reduces both range and maximum height, and the trajectory deviates from a perfect parabola. For most introductory physics problems and educational purposes, the idealized model without air resistance provides an excellent approximation.
How does initial height affect the trajectory?
Increasing the initial height increases both the time of flight and the range of the projectile, but does not affect the maximum height reached above the launch point. The trajectory maintains its parabolic shape, shifted upward by the initial height value.
What is the difference between trajectory and projectile motion?
Trajectory refers specifically to the path the projectile follows through space (the curved line), while projectile motion encompasses the entire study of the projectile's motion including velocity components, time of flight, range, and maximum height. The trajectory is the visual representation of projectile motion.
What is the equation of the trajectory path?
The trajectory equation in standard form is y = h₀ + x·tan(α) − g·x²/(2·v₀²·cos²(α)). This is a quadratic equation in x, confirming the parabolic shape. The coefficient of x² determines how quickly the projectile descends, while the tan(α) term controls the initial slope of the launch.