Projectile Range Calculator

General Physics

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

Practical Examples

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

Cannonball Fired from Ground Level

Cannonball

A cannonball is fired from ground level with a high initial velocity.

Initial Velocity (v₀): 100 m/s, Launch Angle (θ): 30°, Initial Height (y₀): 0 m

Gravity (g): 9.81 m/s²

Golf Ball Drive

Golf Ball

A golf ball is hit with a specific angle and velocity.

Initial Velocity (v₀): 70 m/s, Launch Angle (θ): 15°, Initial Height (y₀): 0 m

Gravity (g): 9.81 m/s²

Rock Thrown from a Cliff

Rock from Cliff

A rock is thrown from a cliff, starting from a significant height.

Initial Velocity (v₀): 20 m/s, Launch Angle (θ): 45°, Initial Height (y₀): 50 m

Gravity (g): 9.81 m/s²

Astronaut Jumping on the Moon

Moon Jump

An astronaut jumps on the moon, where gravity is much lower.

Initial Velocity (v₀): 5 m/s, Launch Angle (θ): 60°, Initial Height (y₀): 0 m

Gravity (g): 1.62 m/s²

Other Titles
Understanding the Projectile Range Calculator: A Comprehensive Guide
Learn the physics behind projectile motion, how to use this calculator effectively, and its real-world applications.

What is Projectile Motion?

  • Defining Projectile Motion
  • Key Components of Trajectory
  • The Role of Gravity
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. This calculator helps you analyze this motion by breaking it down into key metrics.
Defining Projectile Motion
In physics, a projectile is any object that, once projected or dropped, continues in motion by its own inertia and is influenced only by the downward force of gravity. The key assumption is that air resistance is negligible, which simplifies the calculations significantly.
Key Components of Trajectory
The trajectory of a projectile is determined by three main factors: the initial velocity (the speed it starts with), the launch angle (the direction it's thrown), and the initial height. These components dictate how far it travels (range), how high it goes (maximum height), and how long it stays in the air (time of flight).
The Role of Gravity
Gravity is the constant downward force that pulls the projectile towards the center of the Earth (or another celestial body). It causes the vertical velocity of the projectile to change, ultimately bringing it back down. On Earth, the acceleration due to gravity (g) is approximately 9.81 m/s².

Factors not considered in basic model:

  • Air Resistance: In reality, air resistance (drag) significantly affects the trajectory, especially for fast-moving or light objects.
  • Spin: The spin of an object (like a curveball in baseball) can alter its path due to aerodynamic forces (Magnus effect).
  • Earth's Rotation: For very long-range projectiles, the Coriolis effect due to the Earth's rotation becomes a factor.

Step-by-Step Guide to Using the Projectile Range Calculator

  • Inputting Your Values
  • Selecting Gravity
  • Interpreting the Results
Our calculator is designed to be intuitive and easy to use. Follow these steps to get your results.
Inputting Your Values
  1. Initial Velocity (v₀): Enter the launch speed in meters per second (m/s). This must be a positive number.
  2. Launch Angle (θ): Enter the angle of launch in degrees. This value should be between 0° (horizontal) and 90° (vertical).
  3. Initial Height (y₀): Enter the starting height in meters (m). If launching from the ground, this will be 0.
Selecting Gravity
The calculator defaults to Earth's gravity (9.81 m/s²). You can select other celestial bodies like the Moon or Mars from the dropdown to see how different gravitational forces affect the trajectory, or you can input a custom value.
Interpreting the Results

After clicking 'Calculate', you will see three key results:

  • Range: The total horizontal distance the projectile travels before hitting the ground.
  • Maximum Height: The highest point the projectile reaches in its trajectory relative to the launch point.
  • Time of Flight: The total time the projectile spends in the air.

Mathematical Derivation and Formulas

  • Equations of Motion
  • Calculating Time of Flight
  • Deriving Range and Maximum Height
The calculations are based on fundamental kinematic equations. The initial velocity (v₀) is broken down into horizontal (v₀x) and vertical (v₀y) components:

v₀x = v₀ cos(θ) v₀y = v₀ sin(θ)

Calculating Time of Flight (T)
When the projectile starts and ends at the same height (y₀=0), the time of flight is T = (2 v₀y) / g. When the initial height is a factor, the time is found by solving the quadratic equation for vertical motion: y = y₀ + v₀yt - 0.5gt². The time of flight is the positive root of this equation when y=0.
T = [v₀y + sqrt(v₀y² + 2gy₀)] / g
Deriving Range (R) and Maximum Height (H)
The range is the horizontal distance traveled during the time of flight: R = v₀x T. The maximum height is reached when the vertical velocity becomes zero. It is calculated as H = y₀ + (v₀y²) / (2 g). For a launch from y₀=0, the maximum range is achieved at a launch angle of 45°.

Real-World Applications of Projectile Motion

  • Sports Science
  • Military and Ballistics
  • Engineering and Entertainment
The principles of projectile motion are fundamental to many fields.
Sports Science
Athletes and coaches use these principles to optimize performance. In sports like basketball, shot put, golf, and football, understanding the optimal launch angle and velocity can mean the difference between scoring and missing. For example, a golfer selects a club and adjusts their swing to control the ball's trajectory for distance and accuracy.
Military and Ballistics
Ballistics, the study of firearms and projectiles, relies heavily on these calculations. Firing artillery, mortars, or even a simple arrow requires a precise understanding of its path to hit a target accurately. Factors like initial speed (muzzle velocity) and angle are critical.
Engineering and Entertainment
Engineers use projectile motion principles to design everything from fountain water jets to rollercoaster paths. In filmmaking and video games, realistic physics engines simulate projectile trajectories to create believable special effects and gameplay mechanics, such as an explosion sending debris flying or an angry bird being launched from a slingshot.

Common Misconceptions and Correct Methods

  • Heavier Objects Fall Faster
  • A 45° Angle is Always Optimal
  • Horizontal and Vertical Motion are Dependent
Let's clarify some common misunderstandings about projectile motion.
Myth: Heavier Objects Fall Faster
Correction: In the absence of air resistance, all objects fall at the same rate of acceleration (g), regardless of their mass. A cannonball and a feather dropped in a vacuum would hit the ground at the same time. Mass affects momentum and is important when air resistance is considered, but it does not change the acceleration due to gravity itself.
Myth: A 45° Angle is Always Optimal for Maximum Range
Correction: This is only true when the launch and landing heights are the same. If you are launching from a height (like a cliff), the optimal angle for maximum range will be less than 45°. Conversely, if you are launching into a target that is higher than your launch point, the optimal angle will be greater than 45°.
Myth: Horizontal and Vertical Motion are Dependent
Correction: A key principle of projectile motion is the independence of horizontal and vertical motion. Gravity only affects the vertical component of velocity, causing the projectile to accelerate downwards. The horizontal component of velocity remains constant (assuming no air resistance). This is why you can analyze them separately.