Example 1
An object is thrown horizontally from a ledge at 6 m/s, from a height of 8 m, at g = 9.81 m/s². This is the reference example on this page.
Z = 7.66 m
Horizontal Throw Calculator
Enter the horizontal speed and the height of the ledge. The calculator finds the flight time from t = √(2h/g) and the range from Z = v₀·t. The launch angle is 0° and is not a separate field. At 6 m/s from a height of 8 m the time is 1.28 s and the range is 7.66 m. The model does not include air drag.
Result
Enter the horizontal speed and the height. The range, the flight time, and the landing speed will appear here.
You know only the height and want the fall time? Compute t = √(2h/g) in the free-fall time calculator. An angle other than 0°: projectile motion.
How it works
2h/g1.63 s²
t1.28 s
Z7.66 m
A horizontal throw is motion in which the initial velocity is parallel to the ground and the vertical component at launch is zero. The flight time from a height h follows t = √(2h/g), and the range follows Z = v₀·t. In the Discorsi (1638) Galileo described this kind of flight as uniform horizontal motion combined with uniformly accelerated vertical motion. He did not write the constant g in today’s notation. Modern textbooks (Halliday, Resnick, and Walker; Young and Freedman) use the same split into components. For 6 m/s and h = 8 m at g = 9.81 m/s² the ratio 2h/g is 1.63 s², the panel shows a time of 1.28 s, and the range is 7.66 m. The range is computed from the full value of the time, not from the product 6 × 1.28.
The calculator uses the numbers you enter: the horizontal speed v₀, the height h, and the acceleration g. An empty g field means 9.81 m/s². That is the classroom value for Earth, not the standard acceleration 9.80665 m/s². For 6 m/s and 8 m both values show the same range of 7.66 m on the panel, because the result is rounded to two decimal places. A comma and a period are the same decimal separator.
What the result panel shows. Besides the range, the calculator reports the flight time, the vertical speed at landing vᵧ = g·t, the impact speed v = √(v₀² + vᵧ²), the angle of that speed below the horizontal, the ratio 2h/g, the range at double the height, and the entered g. For the reference example the vertical speed is 12.53 m/s (45.1 km/h), the impact speed is 13.89 m/s (50.0 km/h), and the angle is 64.41°. Doubling the height multiplies the time and the range by √2, so from 8 m to 16 m the range grows from 7.66 m to 10.84 m. Four times the height, 32 m, gives a time of 2.55 s and a range of 15.33 m. At 12 m/s from the same 8 m the time stays 1.28 s and the range is 15.33 m. The chart draws the range for heights of 2 m, 4 m, 6 m, 8 m, 12 m, and 20 m at the entered speed and the same g. At 6 m/s and g = 9.81 m/s² those ranges are 3.83 m, 5.42 m, 6.64 m, 7.66 m, 9.38 m, and 12.12 m.
Limits of the model. The formula assumes constant g, no air resistance, and an initial velocity that is exactly horizontal. The angle is not a field. Drag is not subtracted. In air the real range is often shorter, and the flight time is often longer. Mass does not enter the formula. A throw at an angle is computed by the projectile-motion calculator. Range on level ground, from Z = v₀² sin(2α)/g, is on a separate page. At an angle of 0° that formula gives zero, because the landing is at the launch height. Here the landing is lower by h, so the range is not zero. Vertical motion alone is on the vertical-throw calculator. Time from a height, with no horizontal speed, is on the t = √(2h/g) calculator. Distance from a time, starting from rest, is on the h = ½gt² calculator.
The unit switch in the header converts speed to ft/s, height to feet, and g to ft/s², and it converts numbers already entered. Time stays in seconds. Sentences on the page stay in SI units as the reference. The chart axis shows the same launch heights in the unit selected in the header.
The examples use the same formula. A speed of 6 m/s and a height of 8 m give a range of 7.66 m. A speed of 10 m/s from 25 m gives 22.58 m. A speed of 8 m/s from 20 m at g = 1.62 m/s² gives 39.75 m. The button “Fill this example” enters those numbers and refreshes the panel and the chart. The fields start empty.
Formula
t = √(2h/g)
Z = v0 · t
vy = g · t v = √(v0² + vy²) θ = arctan(vy/v0) below the horizontal
Model: a horizontal throw, uniform horizontal motion combined with a vertical drop from rest (Galileo, Discorsi, 1638; the classroom form is t = √(2h/g), Z = v₀·t). The launch angle is 0° and is not an input. The order of calculation is the ratio 2h/g, the square root as the time, then the product v₀·t, then vᵧ, the impact speed, and the angle.
Theory, derivation, and domain
A drop from a ledge
A horizontal throw is the motion of an object that is given a speed only in the horizontal direction and then falls freely in the vertical direction. The height h is the difference between the launch level and the landing level. The horizontal component of velocity stays equal to v₀, because weight has no horizontal component in this model. The vertical component at launch is zero, so the vertical motion is a fall from rest: the vertical distance is h = ½gt². The flight time is therefore t = √(2h/g), and the range is Z = v₀·t.
Independence of the two motions
In the Discorsi (1638) Galileo described a projectile path as uniform horizontal motion combined with naturally accelerated vertical motion. This calculator evaluates the classroom form of that split for an angle of 0°. It does not plot the path point by point. It reports the time, the range, and the velocity at landing.
Why the time ignores v₀
The time is closed by the vertical equation, which does not contain v₀. A larger horizontal speed lengthens only the range. At 6 m/s and 8 m the range is 7.66 m. At 12 m/s from the same height the time is still 1.28 s and the range is 15.33 m. The panel does not multiply the rounded time 1.28 s by the speed. The product 6 × 1.28 would be 7.68 m, and the calculator shows 7.66 m.
Speed and angle at landing
The vertical component at that moment is vᵧ = g·t. For the reference example that is 12.53 m/s. The impact speed joins both components: v = √(v₀² + vᵧ²) = 13.89 m/s, which is 50.0 km/h. The angle of that speed below the horizontal is arctan(vᵧ/v₀) = 64.41°. At v₀ = 0 and a height of 8 m the range is 0.00 m, the time is 1.28 s, and the angle is 90.00°, because the motion is vertical only.
Double height and double speed
The height sits under a square root, so doubling h multiplies the time and the range by √2. From 8 m to 16 m, at 6 m/s, the range grows from 7.66 m to 10.84 m. Four times the height doubles the time before rounding. The panel for 32 m shows 2.55 s and 15.33 m. Doubling the speed at the same h doubles the range and leaves the time unchanged.
What this formula does not compute
The level-ground range Z = v₀² sin(2α)/g describes a launch and a landing at the same height. At α = 0° it gives zero. This calculator launches higher than it lands, so it uses the fall time, not the sine of the doubled angle. There is no field for the angle, the mass, the drag, or a force. An empty g field means exactly 9.81 m/s². The standard acceleration, adopted by the CGPM in 1901, is 9.80665 m/s². The calculator does not switch to that value on its own. For the reference pair, rounding to two decimal places still gives 7.66 m.
How to use
- Enter the horizontal speed in meters per second, for example 6. That is the component along the ledge, not a downward speed.
- Enter the ledge height in meters, for example 8. Zero meters gives a time of 0 s and a range of 0 m.
- Leave g empty for 9.81 m/s², or enter another value, for example 1.62. Do not enter 9.80665 when the problem states 9.81.
- Click Calculate, or wait for the live result. For 6 m/s, 8 m, and an empty g field the calculator shows a range of 7.66 m and a time of 1.28 s.
- Check the panel: the vertical speed, the impact speed, the angle, the ratio 2h/g, and the range at double the height. Clearing the form hides the panel. An angle other than 0° is on a separate page.
Symbol glossary
v₀ is the horizontal speed, h the ledge height, t the flight time, and Z the range. For 6 m/s and 8 m at g = 9.81 m/s² the range is 7.66 m.
- v₀
- Horizontal speed at launch [m/s]. For 6 m/s and 8 m the range is 7.66 m. For 12 m/s from the same height the range is 15.33 m and the time stays 1.28 s. Zero gives a range of 0.00 m.
- h
- Ledge height above the landing level [m]. Zero meters gives a time of 0.00 s. From 32 m at 6 m/s the panel shows 2.55 s and a range of 15.33 m.
- g
- Acceleration [m/s²]. An empty field means 9.81, not 9.80665. It sits under the square root in the flight time.
- t
- Flight time [s], t = √(2h/g). It is not a separate field. For 8 m and an empty g field it is 1.28 s.
- Z
- Range [m], Z = v₀·t, computed from the full value of the time. For the reference example the panel shows 7.66 m.
- vᵧ
- Vertical speed at landing [m/s], vᵧ = g·t. For 8 m it is 12.53 m/s (45.1 km/h).
- v
- Impact speed at landing [m/s], v = √(v₀² + vᵧ²). For 6 m/s and 8 m it is 13.89 m/s (50.0 km/h).
- θ
- Angle of the landing velocity below the horizontal, arctan(vᵧ/v₀). For the reference example it is 64.41°.
Worked examples
Example 2
From a bridge 25 m high, horizontally at 10 m/s. The time is longer than in the reference example because the height is greater. The horizontal speed does not shorten the drop.
Z = 22.58 m
Example 3
From a roof 20 m high, horizontally at 8 m/s, at the Earth value g = 9.81 m/s². Air resistance is not subtracted.
Z = 16.15 m
Example 4
From the edge of a table 3 m high, at 4 m/s. A low edge gives a short flight time.
Z = 3.13 m
Example 5
From a bluff 50 m high, horizontally at 20 m/s. The result is the range in a vacuum, at constant g.
Z = 63.86 m
Example 6
From a window 2.5 m high, horizontally at 3 m/s. The formula is the same as for a high ledge. Only h is smaller.
Z = 2.14 m
Example 7
The same 8 m/s and 20 m as the roof example, but with g = 1.62 m/s² entered from the NASA Moon Fact Sheet. The calculator does not recognize a planet and does not distinguish masses.
Z = 39.75 m
Example 8
From a dock 1.5 m above the water, horizontally at 5 m/s. Air resistance and water resistance are not subtracted. The result is the range to the water surface in this model.
Z = 2.77 m
Common questions
What does this calculator compute, and from which formula?
The calculator finds the flight time of a horizontal throw from t = √(2h/g) and the range from Z = v₀·t. The initial velocity is horizontal, so the vertical component at launch is zero. For 6 m/s and a height of 8 m at g = 9.81 m/s² the panel shows a time of 1.28 s and a range of 7.66 m. The range is computed from the full value of the time, not from the product 6 × 1.28.
Which units should I enter for speed, height, and g?
Enter the speed in meters per second, the height in meters, and g in meters per second squared. The time is in seconds and the range is in meters. After you switch the header to US units, the speed is in ft/s, the height is in feet, and g is in ft/s². Time stays in seconds. A comma and a period are the same decimal separator.
What happens if I leave g empty?
The calculator uses 9.81 m/s². It does not insert the standard acceleration 9.80665 m/s². For 6 m/s and 8 m both values show the same range of 7.66 m on the panel, because the result is rounded to two decimal places. Another value of g, for example 1.62, has to be entered by hand.
Why does the flight time not depend on the horizontal speed?
The flight time comes only from the drop through the height. In the model without drag, the horizontal speed does not shorten or lengthen that time. At 12 m/s from the same 8 m the time is still 1.28 s and the range is 15.33 m. At 6 m/s the range is 7.66 m.
How is this result different from free fall and from an angled throw?
This calculator locks the launch angle at 0° and starts from a height h. Free fall computes a distance from a time with an initial speed of zero, and it does not report a range. A vertical throw has no horizontal component. Projectile motion accepts an angle. The level-ground range Z = v₀² sin(2α)/g gives zero at an angle of 0°, because the landing is at the launch height. Here the landing is lower by h, so the range is not zero.
What do a height of zero and a speed of zero mean?
At a height of 0 m the time is 0.00 s and the range is 0.00 m, because there is no drop. At a speed of 0 m/s and a height of 8 m the range is 0.00 m and the time is 1.28 s, as in a drop from that height. A negative number is rejected. Empty speed and height fields leave the panel without a number.
Does the result include air resistance?
No. Air resistance is a limit of the model, not a separate result. The calculator does not shorten the range or lengthen the time for air. In air the real range is often shorter, and the flight time is often longer. There is no mass field.
Why does doubling the height not double the range?
Because the height sits under a square root. Doubling the height multiplies the time and the range by √2. From 8 m to 16 m, at 6 m/s, the range grows from 7.66 m to 10.84 m. Four times the height, 32 m, gives a time of 2.55 s and a range of 15.33 m. Doubling the speed at the same height doubles the range and leaves the time unchanged.
What do the panel and the chart show?
The panel reports the range, the flight time, the vertical speed at landing, the impact speed, the angle of that speed below the horizontal, the ratio 2h/g, the range at double the height, and the entered g. For 6 m/s and 8 m the vertical speed is 12.53 m/s, the impact speed is 13.89 m/s, and the angle is 64.41°. The chart draws the range for heights of 2 m, 4 m, 6 m, 8 m, 12 m, and 20 m. The button “Fill this example” enters the numbers and refreshes the panel and the chart.
How do I enter a different acceleration?
Enter it in the g field, for example 1.62. For 8 m/s and a height of 20 m the range is then 39.75 m and the time is 4.97 s. At the Earth value 9.81 m/s² the same speed and height give a range of 16.15 m. The calculator does not recognize a planet. It only checks that g is greater than zero.
Do a comma and a period mean the same thing?
Yes. 8.5 and 8,5 are the same height. The calculator accepts both as a decimal separator.
How do I calculate flight time with t = √(2h/g)?
Divide twice the height by g and take the square root. At h = 8 m and g = 9.81 m/s²: √(16/9.81) ≈ 1.28 s.
How do I find range Z at 6 m/s from a height of 8 m?
First the flight time is about 1.28 s, then Z = v₀·t = 6·1.28 ≈ 7.66 m.
Why does flight time not depend on v₀?
Vertically vᵧ₀ = 0, so the fall from height h sets t. Horizontal speed affects only the range Z = v₀·t.
How is a horizontal throw different from free fall?
Flight time matches free fall from that height, but here there is also a constant horizontal component and a range. Free fall has no v₀ field.
How is this page different from an oblique throw?
Here the angle is always 0° and there is no α field. An oblique throw enters an angle plus the components v₀ sin α and v₀ cos α.
Sources
The formula, the split into components, and the units come from mechanics textbooks and from metrology sources. This page does not invent publication identifiers.
- Wikipedia — Projectile motion
- Wikipedia PL — Rzut ukośny
- Wikipedia — Two New Sciences (Galileo, Discorsi, 1638)
- Wikipedia — Standard gravity (9.80665 m/s²)
- BIPM — The International System of Units (SI Brochure)
- NIST — SI units
- NASA — Moon Fact Sheet
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics, chapter on projectile motion.
- Young, H. D.; Freedman, R. A., University Physics, chapter on motion in two dimensions.
Page updated November 3, 2026.