Projectile Motion Calculator

General Physics

This tool helps you analyze the motion of a projectile under gravity. Enter the initial parameters to find the key metrics of its trajectory.

Practical Examples

See how the Projectile Motion Calculator works with real-world scenarios.

Cannonball Fired

Example 1

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

v₀: 100 ms, θ: 30°, y₀: 0 m

Golf Ball Drive

Example 2

A golf ball is hit with an initial velocity of 150 ft/s at an angle of 45 degrees. Assume it starts from the ground.

v₀: 150 fts, θ: 45°, y₀: 0 ft

Stone Thrown from a Cliff

Example 3

A stone is thrown from a 50-meter high cliff with an initial speed of 20 m/s at an angle of 15 degrees above the horizontal.

v₀: 20 ms, θ: 15°, y₀: 50 m

Basketball Shot

Example 4

A basketball is shot from a height of 7 feet, with an initial velocity of 30 ft/s at an angle of 60 degrees.

v₀: 30 fts, θ: 60°, y₀: 7 ft

Other Titles
Understanding Projectile Motion: A Comprehensive Guide
Dive deep into the physics of projectile motion, from basic concepts to complex calculations and real-world applications.

What is Projectile Motion?

  • Defining Projectile Motion
  • Key Assumptions
  • Forces in Action
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The path followed by a projectile is known as its trajectory, which is typically a parabola. Understanding this concept is fundamental in classical mechanics.
Core Principles
The motion is analyzed by separating it into two independent components: horizontal motion and vertical motion. The horizontal component of velocity remains constant throughout the motion (assuming no air resistance), while the vertical component changes due to gravity.
Key Assumptions in Ideal Models
For simplicity, the standard model of projectile motion relies on a few key assumptions: the only force acting on the object is gravity, air resistance is negligible, the rotation of the Earth does not affect the motion, and the acceleration due to gravity (g) is constant.

Step-by-Step Guide to Using the Calculator

  • Input Parameters
  • Unit Selection
  • Interpreting the Results
Our calculator simplifies complex physics into a few easy steps. Here's how to use it effectively.
1. Enter Initial Velocity (v₀)
This is the speed at which the projectile begins its journey. Enter a positive numerical value.
2. Select Velocity Unit
Choose between meters per second (m/s) and feet per second (ft/s). The calculator will adjust gravity accordingly.
3. Enter Launch Angle (θ)
This is the angle of launch relative to the ground, measured in degrees. It must be between 0 (horizontal launch) and 90 (vertical launch).
4. Enter Initial Height (y₀)
This is the starting height from which the projectile is launched. If it starts from the ground, this value is 0.
5. Select Height Unit
Choose between meters (m) and feet (ft). Ensure this is consistent with your velocity unit choice for accurate results.
6. Interpret the Results
After clicking 'Calculate', the tool will display the Time of Flight (how long the projectile is in the air), Maximum Height (the highest point it reaches), and Horizontal Range (the total distance it travels horizontally).

Mathematical Derivation and Formulas

  • Horizontal and Vertical Components
  • Core Kinematic Equations
  • Deriving Key Metrics
The calculations are based on fundamental kinematic equations. Here's a look at the math behind the results.
Component Breakdown
Initial velocity (v₀) is broken down into horizontal (vₓ) and vertical (vᵧ) components: vₓ = v₀ cos(θ) and vᵧ = v₀ sin(θ).
Time of Flight (T)
The time of flight is calculated by solving the vertical motion equation y(t) = y₀ + vᵧt - 0.5gt² for when y(t) = 0 (or the landing height). The full quadratic formula solution gives: T = (vᵧ + √(vᵧ² + 2gy₀)) / g.
Maximum Height (H)
The maximum height is reached when the vertical velocity becomes zero. Using the equation vᵧ² = (v₀sin(θ))² - 2g(H - y₀), we can solve for H: H = y₀ + (v₀sin(θ))² / (2g).
Horizontal Range (R)
Since horizontal velocity is constant, the range is simply the horizontal velocity multiplied by the total time of flight: R = vₓ * T.

Calculation Example

  • Given: v₀ = 50 m/s, θ = 45°, y₀ = 0 m, g = 9.81 m/s².
  • vₓ = 50 * cos(45°) ≈ 35.36 m/s.
  • vᵧ = 50 * sin(45°) ≈ 35.36 m/s.
  • Time to peak = vᵧ / g ≈ 3.6s. Total Time of Flight (T) = 2 * 3.6s = 7.2s.
  • Maximum Height (H) = (vᵧ)² / (2g) ≈ 127.4 m.
  • Horizontal Range (R) = vₓ * T ≈ 254.8 m.

Real-World Applications of Projectile Motion

  • Sports Science
  • Military and Ballistics
  • Engineering and Entertainment
The principles of projectile motion are not just academic; they are applied in numerous fields.
Sports Analysis
In sports like basketball, golf, baseball, and football, understanding the trajectory of a ball is crucial for performance. Athletes and coaches use these principles to optimize shots, throws, and kicks.
Ballistics
In military and forensic science, ballistics is the study of the flight of bullets and other projectiles. Calculating trajectory is essential for aiming and for crime scene reconstruction.
Video Games and Film
Physics engines in video games and special effects in movies rely heavily on accurate projectile motion calculations to create realistic animations and environments.

Common Misconceptions and Correct Methods

  • Mass and Trajectory
  • Gravity's Constant Influence
  • The 45-Degree Angle Myth
Several common misunderstandings exist regarding projectile motion. Let's clarify them.
Does Mass Affect the Path?
In the ideal model (ignoring air resistance), the mass of the projectile does not affect its trajectory. A heavy cannonball and a light tennis ball, launched with the same initial velocity and angle, will follow the same path.
Horizontal and Vertical Motion are Independent
A common mistake is to think that gravity affects horizontal speed. This is incorrect. Gravity only acts vertically, changing the vertical velocity. The horizontal velocity remains constant.
The Myth of the 45-Degree Angle
It's often taught that a 45-degree launch angle gives the maximum range. This is only true when the launch and landing heights are the same (y₀ = 0). If an object is launched from a height, the optimal angle for maximum range will be less than 45 degrees.