Projectile Motion & Trajectory Calculator

General Physics

Enter the initial velocity, launch angle, and initial height to calculate the projectile's trajectory, including maximum height, range, and time of flight.

Practical Examples

Use these examples to see how the calculator works in different scenarios.

Example 1: Cannonball Launch

metric

A cannonball is fired from the ground (0m height) with an initial velocity of 100 m/s at a 30° angle.

v₀: 100, α: 30°, h: 0

Units: metric

Example 2: Baseball Throw

imperial

A baseball is thrown from a height of 6 ft with an initial velocity of 90 ft/s at a 45° angle.

v₀: 90, α: 45°, h: 6

Units: imperial

Example 3: Golf Drive

metric

A golf ball is hit from a tee (0.05m height) with an initial velocity of 70 m/s at a 15° angle.

v₀: 70, α: 15°, h: 0.05

Units: metric

Example 4: Cliff Jump Scenario

imperial

An object is launched horizontally (0° angle) from a 150 ft cliff with a speed of 50 ft/s.

v₀: 50, α: 0°, h: 150

Units: imperial

Other Titles
Understanding Trajectory: A Comprehensive Guide
Explore the physics behind projectile motion, from basic concepts to real-world applications.

What is Projectile Motion?

  • Definition of a Projectile
  • The Role of Gravity
  • Key Assumptions in Calculations
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. This motion is a fundamental concept in classical mechanics and has wide-ranging applications.
The Two Components of Motion
To simplify the analysis, projectile motion is broken down into two independent components: horizontal and vertical. The horizontal motion has a constant velocity, as there are no horizontal forces acting on the projectile (air resistance is ignored). The vertical motion is subject to a constant downward acceleration due to gravity (g), which is approximately 9.81 m/s² or 32.2 ft/s² near the Earth's surface.

Common Projectiles

  • A football being kicked.
  • A bullet fired from a gun.
  • A diver jumping off a platform.

Step-by-Step Guide to Using the Trajectory Calculator

  • Inputting Your Values
  • Selecting Units
  • Interpreting the Results
Our calculator simplifies the process of analyzing projectile motion. Here's how to use it effectively:
1. Enter Initial Parameters
Provide the initial velocity, the launch angle, and the initial height. Ensure the launch angle is between 0 and 90 degrees.
2. Choose Your Units
Select either 'Metric' (meters, m/s) or 'Imperial' (feet, ft/s) units. The calculator will perform all calculations and display results in the chosen system.
3. Analyze the Output
The calculator provides three key outputs: Maximum Height (the peak of the trajectory), Time of Flight (how long the projectile is airborne), and Horizontal Range (the distance it covers).

Real-World Applications of Trajectory Calculation

  • Sports Science
  • Military and Ballistics
  • Engineering and Entertainment
The principles of projectile motion are not just for textbooks; they are applied in numerous fields.
Sports Analytics
In sports like basketball, golf, and football, understanding the trajectory of a ball is crucial for optimizing performance. Athletes and coaches use these principles to perfect a shot, kick, or throw.
Ballistics
In military applications, calculating the trajectory of bullets, missiles, and artillery shells is essential for accuracy. The launch angle and velocity must be precise to hit a target miles away.

Common Misconceptions and Correct Methods

  • The '45-Degree Rule'
  • Gravity's Effect on Horizontal Speed
  • Weight vs. Trajectory
The Optimal Angle Isn't Always 45°
It's a common belief that a 45° launch angle always yields the maximum range. This is only true when the launch and landing heights are the same. If an object is launched from an elevation, the optimal angle will be different.
Gravity and Motion Components
Another misconception is that gravity affects the projectile's horizontal speed. Gravity only acts vertically, changing the vertical velocity. The horizontal velocity remains constant throughout the flight (ignoring air resistance).

Mathematical Derivation and Formulas

  • Equations of Motion
  • Deriving Range and Height
  • The Trajectory Equation
The calculations are based on fundamental kinematic equations:
Core Formulas
Horizontal Position: x = v₀ₓ * t
Vertical Position: y = h + v₀y t - 0.5 g * t²
Horizontal Velocity (v₀ₓ): v₀ * cos(α)
Vertical Velocity (v₀y): v₀ * sin(α)
From these, we derive the formulas for time of flight, maximum height, and range. For instance, the time of flight for a projectile launched from an elevated position h is given by t = (v₀y + √(v₀y² + 2gh)) / g.