Horizontal Projectile Motion Calculator

Our Horizontal Projectile Motion Calculator helps you solve problems involving objects launched horizontally from a certain height. This handy physics tool takes the initial values—like launch height and initial velocity—and calculates everything you need to know about the object’s journey through space.

The beauty of a horizontal projectile motion calculator is that it splits the problem into two simple parts: the horizontal movement (which happens at constant speed) and the vertical movement (which is affected by gravity). This approach makes complicated problems much easier to solve.

Horizontal Projectile Motion Calculator

Imagine you’re standing on a 45-meter cliff and throw a baseball horizontally at 25 meters per second. A horizontal projectile motion calculator would tell you that the ball will:

• Travel approximately 113 meters horizontally before hitting the ground
• Take about 3.03 seconds to reach the ground
• Hit the ground at a speed of roughly 32.7 meters per second

Rather than crunching these numbers manually, the calculator does the heavy lifting for you!

Horizontal Projectile Motion Formula

For the horizontal component (x-direction):

• x = vₓ × t
  • x = horizontal distance traveled
  • vₓ = initial horizontal velocity
  • t = time

For the vertical component (y-direction):

• y = h – ½gt²
  • y = vertical position (usually set to 0 when object hits ground)
  • h = initial height
  • g = acceleration due to gravity (9.8 m/s²)
  • t = time

Additional useful formulas include:

• Time to hit ground: t = √(2h/g)
• Range (horizontal distance): R = vₓ × √(2h/g)
• Final velocity: v = √(vₓ² + (gt)²)

The horizontal projectile motion formulas are derived from the fundamental equations of motion. What makes these equations special is that the horizontal and vertical components are treated independently:

Example: Let’s calculate how far a ball will travel if thrown horizontally at 15 m/s from a 20-meter tall building:

1. Calculate time to hit ground: t = √(2 × 20/9.8) = √(40/9.8) = √4.08 = 2.02 seconds
2. Calculate horizontal distance: x = 15 × 2.02 = 30.3 meters

This means the ball will land about 30.3 meters away from the building’s base.

How to Calculate Horizontal Projectile Motion

Calculating horizontal projectile motion involves these straightforward steps:

• Identify the known values:
  • Initial height (h)
  • Initial horizontal velocity (vₓ)
  • Acceleration due to gravity (g = 9.8 m/s²)
• Calculate the time of flight:
  • Use the formula t = √(2h/g)
  • This tells you how long the object is in the air
• Determine the horizontal distance:
  • Use the formula x = vₓ × t
  • This gives you the range of the projectile
• Calculate the final velocity (optional):
  • Use the formula v = √(vₓ² + (gt)²)
  • This provides the speed at impact
• Find the angle of impact (optional):
  • Use θ = tan⁻¹(gt/vₓ)
  • This gives you the angle at which the object hits the ground

A water balloon is launched horizontally at 12 m/s from a window 15 meters above the ground. Let’s calculate its motion:

• Time to hit ground: t = √(2 × 15/9.8) = √(30/9.8) = √3.06 = 1.75 seconds
• Horizontal distance traveled: x = 12 × 1.75 = 21 meters
• Final velocity: v = √(12² + (9.8 × 1.75)²) = √(144 + 294.01) = √438.01 = 20.93 m/s
• Angle of impact: θ = tan⁻¹((9.8 × 1.75)/12) = tan⁻¹(14.29/12) = tan⁻¹(1.19) = 50°

So the water balloon will travel 21 meters horizontally, hit the ground at 20.93 m/s, and impact at a 50-degree angle.

What is Horizontal Projectile Motion?

Horizontal projectile motion is a special case of projectile motion where an object is launched purely in the horizontal direction (with no initial vertical velocity) and then follows a curved path due to gravity’s influence.

This type of motion appears in countless real-world scenarios:

• A ball rolling off a table
• Water flowing from a horizontal pipe
• A package dropped from a moving airplane
• A bullet fired horizontally from a gun
• A long jumper leaping from a platform

Example 1: Soccer Ball Kicked Off a Cliff

A soccer ball is kicked horizontally at 18 m/s from the edge of a 50-meter cliff.

• Time to hit ground: t = √(2 × 50/9.8) = √(100/9.8) = √10.2 = 3.19 seconds
• Horizontal distance: x = 18 × 3.19 = 57.42 meters
• Final velocity: v = √(18² + (9.8 × 3.19)²) = √(324 + 976.75) = √1300.75 = 36.07 m/s

The soccer ball lands 57.42 meters away from the cliff base at a speed of 36.07 m/s.

Example 2: Marble Rolling Off a Lab Table

A marble rolls off a 0.8-meter high lab table with a horizontal velocity of 2 m/s.

• Time to hit ground: t = √(2 × 0.8/9.8) = √(1.6/9.8) = √0.163 = 0.404 seconds
• Horizontal distance: x = 2 × 0.404 = 0.808 meters
• Final velocity: v = √(2² + (9.8 × 0.404)²) = √(4 + 15.69) = √19.69 = 4.44 m/s

The marble lands 0.808 meters (about 81 cm) from the table’s edge at a speed of 4.44 m/s.

Example 3: Water Stream from a Hose

Water exits a horizontally-aimed hose at 5 m/s from a height of 1.2 meters.

• Time to hit ground: t = √(2 × 1.2/9.8) = √(2.4/9.8) = √0.245 = 0.495 seconds
• Horizontal distance: x = 5 × 0.495 = 2.475 meters
• Final velocity: v = √(5² + (9.8 × 0.495)²) = √(25 + 23.52) = √48.52 = 6.97 m/s

The water stream will hit the ground 2.475 meters away from the hose at a speed of 6.97 m/s.

References

• Physics Classroom – Projectile Motion – Detailed tutorials with animations explaining horizontal and vertical components.
• HyperPhysics (Georgia State University) – Academic resource with detailed explanations of trajectories and projectile motion concepts.