Calculation
t = √(2h/g); x = v0·t
Vertical fall and horizontal motion are independent; fall time determines how far the body travels horizontally.
x = v0·√(2h/g)
Horizontal projectile motion: range
Vertical fall and horizontal motion are independent; fall time determines how far the body travels horizontally.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Range x — Horizontal distance travelled from launch to impact on the level below the launch point.
- Launch height h — Vertical height of the launch point above the landing surface.
- Horizontal initial speed v0 — Constant horizontal velocity component at launch, with no vertical component.
- Gravitational acceleration g — Local gravitational acceleration, about 9.81 m/s² near sea level.
Example
h=1.25 m and v0=3 m/s give a fall time t=√(2·1.25/9.80665)≈0.505 s and a range x≈1.515 m; impact speed is √(v0²+(g·t)²)≈5.90 m/s.
Assumptions and limits
Idealised point mass with no air resistance, constant g and a flat landing surface at the base of height h below launch.
Understand Horizontal projectile motion: range
This calculator determines how far a horizontally launched body travels before it hits a lower level. It uses the fact that vertical fall and horizontal motion in horizontal projectile motion superimpose completely independently.
What does this quantity describe?
Vertical motion corresponds to free fall from height h: fall time follows from h=½·g·t² as t=√(2h/g). Since horizontal speed v0 stays constant throughout the flight, range follows as x=v0·t=v0·√(2h/g).
Formula and variables
t = √(2h/g); x = v0·t
t = √(2h/g)x = v0·tvimpact = √(v0²+(g·t)²)
| Symbol / input | Meaning |
|---|---|
| Range x | Horizontal distance travelled from launch to impact on the level below the launch point. |
| Launch height h | Vertical height of the launch point above the landing surface. |
| Horizontal initial speed v0 | Constant horizontal velocity component at launch, with no vertical component. |
| Gravitational acceleration g | Local gravitational acceleration, about 9.81 m/s² near sea level. |
Choose the inputs correctly
h is the vertical height of the launch point above the landing surface. v0 is the horizontal initial speed at launch; any vertical initial speed component is not considered here. g is gravitational acceleration, defaulting to 9.80665 m/s².
How to use the calculator
Measure h as the actual height difference between launch point and landing surface, not as distance travelled along a slanted path. Use v0 as a purely horizontal velocity component, e.g. from a prior acceleration or speed calculation.
Worked example
h=1.25 m and v0=3 m/s give a fall time t=√(2·1.25/9.80665)≈0.505 s and range x≈1.515 m; impact speed is √(v0²+(g·t)²)≈5.90 m/s.
Understand the result and units
Range grows linearly with v0 but only with the square root of h: quadrupling launch height only doubles range, whereas doubling v0 directly doubles it. Fall time depends solely on h and g, not on v0.
h is a length, v0 a speed and g an acceleration. Range x is output as a length.
Where the relation comes from
Superimposing free fall and uniform horizontal motion is a standard relation of classical mechanics, as derived for example in OpenStax College Physics for horizontal projectile motion. No normative or manufacturer-specific data is used.
Typical applications
Estimating throw distances for conveyor discharge, jumping equipment, chutes with a horizontal exit, and introductory physics teaching to illustrate superimposed motion.
Assumptions, limits and common mistakes
Idealised point mass with no air resistance, constant g over the whole trajectory, and a flat landing surface exactly at height difference h below launch. Rotation, drag, wind and a sloped or uneven landing surface are excluded.
Common mistake: Do not enter a downward-angled initial velocity as a purely horizontal v0. Do not confuse the distance travelled along the curved path with the vertical height h.
Frequently asked questions
Does the formula also apply to an angled throw?
No. This calculator assumes a purely horizontal initial speed with no vertical component; for an angled throw, initial speed and angle would first need to be resolved into horizontal and vertical components.
Why doesn't fall time depend on v0?
Because vertical and horizontal motion superimpose independently: fall time is determined solely by the vertical fall from height h.
Can this also give impact speed?
Impact speed follows from the vector sum of v0 and the vertical fall speed g·t reached at landing; it is stated in the example text but is not itself a selectable target of this calculator.