Projectile Motion Calculator

Calculate range, maximum height, and flight time

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

Examples

  • v₀ = 20 m/s, θ = 45°, g = 9.81 m/s² → Range = 40.77 m, Max Height = 10.20 m, Time = 2.88 s
  • v₀ = 30 m/s, θ = 60°, g = 9.81 m/s² → Range = 79.49 m, Max Height = 34.42 m, Time = 5.30 s
  • v₀ = 15 m/s, θ = 30°, g = 9.81 m/s² → Range = 19.87 m, Max Height = 2.87 m, Time = 1.53 s
Other Titles
Understanding Projectile Motion: A Comprehensive Guide
Explore the physics of projectile motion, equations, and real-world applications.

What is Projectile Motion?

  • Describes the motion of an object thrown into the air
  • Follows a curved, parabolic trajectory
  • Affected by gravity and initial velocity
Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone (assuming air resistance is negligible). The path followed is called a trajectory and is typically a parabola.
Key parameters include initial velocity (v₀), launch angle (θ), and gravitational acceleration (g). These determine the range, maximum height, and time of flight.
Projectile motion is a fundamental concept in physics, with applications in sports, engineering, and nature.

Examples

  • A soccer ball kicked at an angle follows a parabolic path.
  • A cannonball fired from a cannon is a classic example of projectile motion.

How to Use the Projectile Motion Calculator

  • Input initial velocity, angle, and gravity
  • Units must be consistent for all values
  • Interpret the calculated results
To use this calculator, enter the initial velocity (v₀), launch angle (θ), and gravity (g). The calculator will compute the range, maximum height, and time of flight using standard projectile motion equations.
Equations Used:
  • Range: R = (v₀² × sin(2θ)) / g
  • Maximum Height: H = (v₀² × sin²θ) / (2g)
  • Time of Flight: T = (2 × v₀ × sinθ) / g
Tips for Accurate Calculations:
  • Ensure all values are in compatible units (e.g., m/s for velocity, degrees for angle, m/s² for gravity).
  • The standard value for gravity on Earth is 9.81 m/s².
  • The launch and landing heights are assumed to be the same.

Usage Examples

  • To calculate for v₀ = 20 m/s, θ = 45°, g = 9.81 m/s²: Range = 40.77 m, Max Height = 10.20 m, Time = 2.88 s
  • If you increase the angle to 60°, the range decreases but the maximum height increases.

Detailed Guide to Using the Projectile Motion Calculator

  • Follow these steps for accurate calculations
  • Understand the meaning of velocity, angle, and gravity values
  • Learn how to interpret the results
  • Discover tips for avoiding common errors
Our projectile motion calculator is designed to provide quick and accurate calculations for any initial velocity, angle, and gravity. Follow these detailed instructions to get the most out of this tool:
Understanding the Inputs:
  • Initial Velocity (v₀): The speed at which the object is launched. Can be any positive real number.
  • Launch Angle (θ): The angle above the horizontal. Must be between 0° and 90°.
  • Gravity (g): The acceleration due to gravity. Standard value is 9.81 m/s², but can be changed for other planets.
Advanced Usage Tips:
  • For maximum range, use a 45° launch angle on level ground.
  • Increasing gravity decreases both range and maximum height.
  • The calculator assumes launch and landing heights are equal.

Usage Examples

  • To calculate for v₀ = 25 m/s, θ = 30°, g = 9.81 m/s²: Range = 55.19 m, Max Height = 7.97 m, Time = 2.55 s
  • If you use g = 3.7 m/s² (Mars), the same launch gives a much longer range and time.

Real-World Applications of Projectile Motion

  • Sports: Ballistics, basketball, soccer, golf, etc.
  • Engineering: Fireworks, rockets, and artillery
  • Nature: Animal jumps, seeds dispersal, volcanic eruptions
Projectile motion is not just a theoretical concept but is widely used in real-world applications across various fields:
Sports and Games:
  • Calculating the best angle and speed for throwing, kicking, or hitting a ball in sports like soccer, basketball, golf, and baseball.
Engineering and Technology:
  • Designing the trajectories of fireworks, rockets, and artillery shells.
  • Ballistics calculations for military and law enforcement.
Nature and Biology:
  • Analyzing the jumps of animals, the flight of seeds, or the path of volcanic ejecta.

Applications

  • A soccer player uses projectile motion to score a goal from a distance.
  • Engineers use projectile equations to design safe fireworks displays.
  • Biologists study how frogs optimize their jump angles for maximum distance.