Projectile Motion Calculator

General Physics

This tool calculates the maximum height, flight duration, and range of a projectile based on its initial velocity, launch angle, and initial height.

Practical Examples

Explore these common scenarios to see how the calculator works.

Standard Cannonball Shot

example1

A cannonball is fired from the ground at a 45-degree angle.

v₀: 100, θ: 45°, y₀: 0

Units: Metric

Baseball Throw

example2

A player throws a baseball from shoulder height.

v₀: 90, θ: 35°, y₀: 5.5

Units: Imperial

High Arc Shot

example3

A projectile launched at a high angle to maximize height over a short distance.

v₀: 50, θ: 75°, y₀: 1

Units: Metric

Arrow Fired from a Castle Wall

example4

An archer on a castle wall fires an arrow horizontally.

v₀: 150, θ: 10°, y₀: 50

Units: Imperial

Other Titles
Understanding Projectile Motion: A Comprehensive Guide
This guide covers the fundamental principles of projectile motion, from the core formulas to practical applications, helping you understand the physics behind the numbers.

What is Projectile Motion?

  • Defining Projectile Motion
  • The Role of Gravity
  • Key Components of Motion
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. Such an object is called a projectile, and its path is called its trajectory. This motion is two-dimensional, with both horizontal and vertical components.
The Two Components of Motion
The key to analyzing projectile motion is to treat it as two independent motions: horizontal motion and vertical motion. The horizontal motion is constant velocity (assuming no air resistance), while the vertical motion is constant acceleration due to gravity.

Key Terms

  • Initial Velocity (v₀): The speed and direction the object starts with.
  • Launch Angle (θ): The angle at which the object is projected.
  • Trajectory: The curved path the object follows.

Step-by-Step Guide to Using the Maximum Height Calculator

  • Inputting Your Data
  • Selecting the Right Units
  • Interpreting the Results
How to Use the Calculator
  1. Initial Velocity: Enter the speed at which your projectile is launched. This must be a positive number.
  2. Launch Angle: Input the angle of launch in degrees, between 0 and 90.
  3. Initial Height: Enter the starting height. If launching from the ground, this will be 0.
  4. Unit System: Choose 'Metric' (m, m/s) or 'Imperial' (ft, ft/s). The force of gravity will be adjusted automatically.
  5. Calculate: Click the 'Calculate' button to see the results.

Result Fields

  • Maximum Height (H): The highest point the projectile reaches.
  • Time to Max Height: The time it takes to reach the maximum height.
  • Total Flight Time: The total duration the projectile is in the air.
  • Range (R): The total horizontal distance the projectile travels.

Real-World Applications of Projectile Motion

  • Sports Science
  • Military and Ballistics
  • Engineering and Design
The principles of projectile motion are fundamental in many fields. In sports like basketball, baseball, and golf, understanding the trajectory of a ball is crucial for performance. Athletes intuitively use these principles to score points or hit targets.
Ballistics and Engineering
In military applications, ballistics is the science of projectile flight, essential for accurately firing artillery. Engineers use these calculations to design everything from water fountains to the trajectories of space probes.

Application Examples

  • Calculating the path of a soccer ball to score a goal.
  • Determining the trajectory of a firework.
  • Designing a ski jump to ensure a safe landing area.

Common Misconceptions and Correct Methods

  • Ignoring Air Resistance
  • Mass and its Effect
  • Horizontal and Vertical Independence
The Myth of Mass
A common misconception is that a heavier object will fall faster or travel a shorter distance. In the absence of air resistance, the mass of a projectile does not affect its motion. Two objects of different masses, launched with the same initial velocity and angle, will travel the same path.
Air Resistance
This calculator, like most introductory physics models, ignores air resistance. In the real world, air resistance (or drag) can significantly affect the trajectory, especially for fast-moving or lightweight objects. For high-precision calculations, advanced models incorporating drag are necessary.

Key Takeaways

  • Mass does not influence projectile trajectory (in a vacuum).
  • Air resistance is a real-world factor that this simplified model ignores.
  • The horizontal and vertical components of motion are independent of each other.

Mathematical Derivation and Formulas

  • The Core Equations
  • Deriving Maximum Height
  • Calculating Flight Time and Range
Core Formulas

The motion of a projectile is governed by the following equations, where g is the acceleration due to gravity:

  • Horizontal position: x(t) = (v₀ cos(θ)) t
  • Vertical position: y(t) = y₀ + (v₀ sin(θ)) t - 0.5 g t²
  • Vertical velocity: v_y(t) = v₀ sin(θ) - g t
Finding the Maximum Height
The maximum height is reached when the vertical component of velocity (vy) is zero. We solve vy(t) = 0 for t to find the time to max height: t = (v₀ * sin(θ)) / g. We then plug this time back into the vertical position equation to find the maximum height H.

Key Formulas Used

  • Maximum Height (H) = y₀ + (v₀² * sin²(θ)) / (2g)
  • Total Flight Time (T) = [v₀*sin(θ) + sqrt((v₀*sin(θ))² + 2*g*y₀)] / g
  • Range (R) = (v₀ * cos(θ)) * T