Projectile Motion Calculator - Range, Height & Time

Initial Velocity (m/s)

Launch Angle (°)

45° gives the maximum range for a given launch speed on level ground.

Horizontal Range

0 m

Max Height

0 m

Time of Flight

0 s

What is a Projectile Motion?

The Projectile Motion Calculator applies standard kinematics equations to compute the full trajectory of a launched object, its maximum height, time of flight, and horizontal range, from just an initial velocity and launch angle. It assumes standard Earth gravity (9.8 m/s²) and no air resistance, the same idealized model used in introductory physics.

Enter an initial velocity and a launch angle, and the calculator instantly returns all three key trajectory outputs along with a visual diagram showing the parabolic path, launch point, peak, and landing point.

If you need the launch velocity's associated energy, use the Kinetic Energy Calculator; for purely vertical motion without a launch angle, use the Free Fall Calculator instead.

Why Use a Projectile Motion Calculator?

Manually solving projectile motion problems requires breaking velocity into horizontal and vertical components, then applying three separate kinematic equations, a process that's easy to get wrong, especially with trigonometric functions involved.

This calculator does all three calculations at once and updates live as you adjust velocity or angle, making it easy to explore how launch angle affects range, height, and flight time independently, for example, seeing that 45° maximizes range while steeper angles favor height.

The included trajectory diagram gives an intuitive visual check that complements the exact numeric outputs, useful for verifying that a result "looks right" at a glance.

Who Should Use This Calculator?

Physics students solving projectile motion problems and verifying the relationships between launch angle, range, height, and time of flight.

Sports scientists and coaches estimating the trajectory of thrown or kicked objects, such as balls in various sports.

Teachers demonstrating how launch angle affects range and height, including the classic result that 45° maximizes range.

Engineers and hobbyists working on model rockets, catapults, trebuchets, or similar launch-based projects who need quick trajectory estimates.

Game developers implementing simplified physics for projectile-based mechanics who want to verify their formulas against standard kinematics.

What Insights Does This Calculator Give You?

Horizontal Range, the primary result, showing how far the projectile travels horizontally before landing at the same height it launched from.

Maximum Height, the peak vertical height the projectile reaches during its flight, which increases with steeper launch angles.

Time of Flight, the total time the projectile spends in the air, from launch to landing.

Trajectory Shape, the visual diagram shows the parabolic path at a glance, making it easy to compare how different launch angles change the shape of the trajectory even when the range is similar.

How to use this Projectile Motion calculator

Enter the initial velocity, the launch speed of the projectile, in meters per second.
Enter the launch angle, the angle above the horizontal at which the projectile is launched, in degrees (0–90°).
Read the range result, the highlighted result shows the horizontal range in meters.
Check maximum height and time of flight, the two secondary results show the peak height and total flight duration.
View the trajectory diagram, the diagram visualizes the parabolic path, launch point, peak, and landing point based on your inputs.
Adjust the angle to explore range vs. height tradeoffs, try 45° for maximum range, or higher angles to see height increase at the cost of range.

Show formula & methodology ↓Show less ↑

Formula & Methodology

Component velocities:
vₓ = v × cos(θ), v_y = v × sin(θ)

Time of flight:
t = 2 × v_y ÷ g

Maximum height:
h = v_y² ÷ (2 × g)

Horizontal range:
R = vₓ × t

Variable definitions:
- v, initial velocity (meters per second)
- θ, launch angle (degrees)
- g, gravitational acceleration, fixed at 9.8 m/s²
- t, time of flight (seconds)
- h, maximum height (meters)
- R, horizontal range (meters)

Worked example:

A projectile is launched at 25 m/s at a 40° angle.

Step 1, Component velocities: vₓ = 25 × cos(40°) ≈ 19.15 m/s, v_y = 25 × sin(40°) ≈ 16.07 m/s

Step 2, Time of flight: t = 2 × 16.07 ÷ 9.8 ≈ 3.28 s

Step 3, Maximum height: h = 16.07² ÷ (2 × 9.8) ≈ 13.19 m

Step 4, Horizontal range: R = 19.15 × 3.28 ≈ 62.8 m

Note: This calculator assumes launch and landing occur at the same height and ignores air resistance. Real-world trajectories affected by drag, wind, or spin (such as a curveball) will deviate from these idealized results.

Frequently Asked Questions

What formulas does this projectile motion calculator use?

This calculator uses standard projectile motion equations assuming no air resistance: time of flight = 2v·sin(θ)/g, maximum height = v²·sin²(θ)/(2g), and horizontal range = v²·sin(2θ)/g, where v is initial velocity, θ is launch angle, and g is gravitational acceleration (9.8 m/s²).

What launch angle gives the maximum range?

For a projectile launched and landing at the same height, 45° gives the maximum horizontal range for any given launch velocity. Angles above or below 45° (like 30° or 60°) produce equal ranges to each other but less than the 45° maximum, while affecting time of flight and maximum height differently.

Does this calculator account for air resistance?

No, this calculator uses idealized projectile motion equations that ignore air resistance (drag), which is the standard simplification used in introductory physics. Real-world projectiles, especially lightweight or high-speed ones, experience some deceleration from air resistance that these equations don't capture.

Why does this calculator assume the projectile lands at the same height it launched from?

The standard equations used here assume launch and landing occur at the same vertical height (like a ball thrown and caught at the same level), which is the most common textbook scenario. If launch and landing heights differ, the time of flight and range calculations require additional terms not included in this simplified model.

What is the relationship between launch angle and maximum height?

Maximum height increases as the launch angle increases toward 90° (straight up), since more of the initial velocity is directed vertically. At 90°, the projectile has no horizontal range at all (it goes straight up and comes straight back down), while at 0° it has no height gain (it moves purely horizontally).

How is time of flight affected by launch angle?

Time of flight increases as the launch angle increases, because a steeper angle sends more of the initial velocity into the vertical component, which takes longer to fall back down under gravity. A projectile launched straight up (90°) has the longest possible flight time for a given velocity, though it has zero range.

How can I find the launch velocity needed for a certain kinetic energy?

Use the [Kinetic Energy Calculator](/kinetic-energy-calculator/) with the projectile's mass and its initial velocity (the value you enter here) to find its kinetic energy at launch, useful for understanding the energy requirements of a given projectile motion scenario.

What is a typical use case for this calculator?

This calculator is commonly used for physics homework involving thrown or launched objects (balls, projectiles, water from a hose), estimating the range of sports equipment (like a kicked or thrown ball), and basic ballistics or engineering estimates where air resistance can be reasonably ignored.

How does gravity affect the results?

This calculator uses Earth's standard gravitational acceleration of 9.8 m/s². A lower gravity (like the Moon's 1.62 m/s²) would produce a much longer time of flight, greater maximum height, and greater range for the same launch velocity and angle, since less force pulls the projectile back down.

How is the trajectory diagram generated?

The trajectory diagram plots a parabolic path using the calculated maximum height and horizontal range, showing the launch point, peak height, and landing point to scale, giving a quick visual sense of the projectile's path in addition to the exact numeric outputs.

What's the difference between this calculator and the Free Fall Calculator?

The [Free Fall Calculator](/free-fall-calculator/) handles purely vertical motion starting from rest (like dropping an object), while this Projectile Motion Calculator handles motion with both horizontal and vertical velocity components from a launch angle, producing range and trajectory shape in addition to fall time.

Can this calculator be used for launch angles greater than 90°?

No, launch angles are limited to between 0° and 90°, since these represent all physically meaningful directions of upward-and-forward motion for a projectile launched from ground level. Angles beyond 90° would represent launching backward or downward, which fall outside this calculator's standard forward-projectile model.

Also known as

trajectory calculatorrange of projectile calculatorlaunch angle calculatorprojectile range and height calculatortime of flight calculator