Calculate Projectile Range
Enter consistent values. The answer appears above this form after calculation.
Example Data Table
| Mode | Speed | Angle | Height | Gravity | Range |
|---|---|---|---|---|---|
| Level | 20 m/s | 45° | 0 m | 9.80665 m/s² | 40.79 m |
| Elevated | 20 m/s | 35° | 10 m | 9.80665 m/s² | 49.41 m |
| Elevated | 30 m/s | 25° | 5 m | 9.80665 m/s² | 79.76 m |
Formula Used
For equal launch and landing heights, the calculator uses the level-ground projectile equation.
R = v² × sin(2θ) ÷ g
R is horizontal range. v is launch speed. θ is the angle above horizontal. g is gravitational acceleration.
For a raised launch, it first calculates the positive flight time. It then uses horizontal velocity and time.
t = [v sin(θ) + √((v sin(θ))² + 2gh)] ÷ gR = v cos(θ) × t
h is the initial height above the landing level. These equations assume no air resistance and constant gravity.
How to Use This Calculator
- Choose level mode when launch and landing heights are equal.
- Choose elevated mode when the object starts above ground level.
- Select metric or imperial units before entering values.
- Enter launch speed, angle, gravity, and initial height when needed.
- Choose your preferred decimal precision and submit the form.
- Review the range, flight time, velocity components, and displayed formula.
- Download the result as CSV or PDF when you need a record.
Understanding Projectile Range
Projectile range is the horizontal distance between launch and landing. A range calculation joins horizontal motion with vertical motion. The standard model ignores air resistance. It also treats gravity as constant. These assumptions make the flight path a parabola.
For level launch and landing points, the calculator uses R = v² sin(2θ) / g. Here, v is launch speed, θ is launch angle, and g is gravitational acceleration. A 45-degree angle gives maximum level-ground range at a fixed speed. Complementary angles, such as 30 and 60 degrees, give equal range on level ground.
A raised launch point needs a different method. The projectile stays airborne longer. The calculator finds flight time first. It then multiplies that time by horizontal speed. This suits ramps, platforms, cliffs, and elevated launchers. It handles a horizontal launch from a raised position.
Units and Input Choices
Use one consistent length system for every field. Pair metres with metres per second and metres per second squared. Pair feet with feet per second and feet per second squared. Mixed units create false answers. Earth gravity is about 9.80665 metres per second squared. It is about 32.174 feet per second squared.
Measure the angle above the horizontal. A shallow angle provides more horizontal speed. A steep angle provides more vertical speed. Initial height also changes the final distance. Extra height gives more falling time. That time may increase horizontal travel.
The calculator displays horizontal and vertical velocity components. Horizontal velocity stays constant in the ideal model. Vertical velocity changes because gravity acts downward. Flight time shows how long the object remains in the air.
Reading the Result
Range is measured horizontally. It is not the curved distance travelled through the air. The result assumes a flat landing surface. In elevated mode, ground level is below the launch point. Confirm that this matches your question before using the number.
Rounding only changes the display. It does not change the main calculation. Choose more decimal places for laboratory work. Choose fewer decimals for classroom estimates. The displayed equation mode helps checking.
Change inputs one at a time. Increase speed to see its effect. Adjust angle to compare flight shapes. Change gravity for another planet or experiment. Record units and assumptions. Clear notes make results easier to review.
Real-World Limits
The ideal range equation is useful, but it has limits. Air drag slows a projectile. Wind can push it sideways. Spin can change lift and path shape. Uneven terrain changes the landing level. These conditions need a more detailed model.
Use this calculator for learning, planning, and textbook checks. Do not use it alone for safety-critical launches. Measure inputs. Record the launch point and landing level. Compare estimates with observed results when possible.
A sound calculation starts with clear inputs. It ends with a check. Range should fit the speed, angle, and environment. Review outputs before relying on them. Consistent units and realistic assumptions produce dependable projectile range estimates.
Frequently Asked Questions
What does projectile range mean?
Projectile range is the horizontal distance from the launch point to the landing point. This calculator reports that distance in metres or feet, based on your selected unit system.
Which formula is used on level ground?
For equal launch and landing heights, the calculator uses R = v² sin(2θ) / g. This assumes constant gravity, no air resistance, and a flat landing surface.
Why is the elevated formula different?
A raised launch remains airborne longer. The calculator solves the vertical-motion equation for time first, then multiplies time by horizontal velocity to calculate range.
Can I enter a zero-degree angle?
Yes. A zero-degree angle gives zero range on level ground. From an elevated point, it models a horizontal launch and can still produce a positive range.
What angle gives maximum range?
On level ground with no air resistance, 45 degrees gives maximum range at a fixed launch speed. A different launch height can shift the best angle.
Can I use feet instead of metres?
Yes. Select the imperial option, enter speed in feet per second, height in feet, and gravity in feet per second squared. Keep every value consistent.
Does the calculator include air resistance?
No. It uses an ideal projectile model. Drag, wind, spin, and changing terrain can change actual range. Use experimental data or advanced simulation when those effects matter.
What is flight time?
Flight time is the duration from launch until the projectile reaches the landing level. It helps explain why speed, angle, height, and gravity affect horizontal distance.
Why does initial height increase range?
Greater initial height usually gives extra time before landing. With positive horizontal velocity, that additional time lets the projectile travel farther across the ground.
How many decimal places should I use?
Use two or three decimals for common problems. Use more for experiments or precise comparisons. Avoid reporting more precision than your measured input values justify.
How can I avoid range errors?
Check the angle, unit system, gravity, and landing-height assumption. Use consistent units for dependable projectile range calculations always.