Example 1
A ball goes straight up at 15 m/s, at g = 9.81 m/s². This is the reference example on this page. No time is entered.
H_max = 11.47 m
Vertical Throw Calculator
Choose an upward or a downward throw and enter the initial speed. Upward, the calculator computes H_max = v₀²/(2g) and the climb time v₀/g. At 15 m/s the height is 11.47 m and the time to the peak is 1.53 s. The model does not include air drag or a horizontal range.
Result
Enter the initial speed. Maximum height, the times, and the state at time t will appear here.
You know only the height and want the fall time from rest? Compute t = √(2h/g) in the free-fall time calculator. A horizontal speed off an edge: horizontal throw.
How it works
v₀²225 (m/s)²
H_max11.47 m
t up1.53 s
A vertical throw is motion along one axis, with constant acceleration g and no horizontal component. In the upward mode the axis points up, so speed and height follow v(t) = v₀ − g t and h(t) = v₀ t − ½ g t². At the peak the speed is zero, so the maximum height follows H_max = v₀²/(2g) and the climb time follows t_up = v₀/g. The return to launch level takes T = 2 v₀/g. In the Discorsi (1638) Galileo described naturally accelerated motion. He did not write the constant g in today’s notation. Textbooks (Halliday, Resnick, and Walker; Young and Freedman) give the same school form. For 15 m/s at g = 9.81 m/s² the quotient v₀²/(2g) is 11.47 m, the panel shows a climb of 1.53 s, and the flight time is 3.06 s. The product 15 × 1.53 / 2 is 11.475, which rounds to 11.48 m at two decimal places. The panel shows 11.47 m because it uses the full quotient.
The calculator uses the numbers you enter: the direction, the initial speed v₀, an optional time t, and the acceleration g. An empty g field means 9.81 m/s². That is the school value for Earth, not the standard acceleration 9.80665 m/s². For 15 m/s both values show the same height of 11.47 m and the same time of 1.53 s on the panel, because the result is rounded to two decimal places. A comma and a period are the same separator. 12.5 and 12,5 give a height of 7.96 m and a climb time of 1.27 s.
What the result panel shows. Besides the maximum height, the calculator reports the time to the peak and the flight time, the state v(t) and h(t) when a time is entered, the entered speed, the g in use, and the speed on return to launch level. That speed equals v₀, because the return is symmetric when there is no drag. For 15 m/s it is 15.00 m/s (54.0 km/h). For 10 m/s after 1 s the speed is 0.19 m/s (0.7 km/h) and the height is 5.10 m. The climb at 10 m/s lasts 1.02 s, so the instant 1 s is still before the peak. The maximum height at 10 m/s also displays as 5.10 m. Before rounding, those two heights differ by just under 2 mm. Doubling the speed from 15 m/s to 30 m/s gives a height of 45.87 m, because the formula squares the speed. The climb time grows to 3.06 s. For an upward throw with v₀ greater than zero, the chart draws the height at six instants: 0%, 20%, 40%, 60%, 80%, and 100% of the flight time. For 15 m/s those instants are 0.00 s, 0.61 s, 1.22 s, 1.83 s, 2.45 s, and 3.06 s, and the heights are 0.00 m, 7.34 m, 11.01 m, 11.01 m, 7.34 m, and 0.00 m.
Limits of the model. The formula assumes constant g, no air resistance, and a velocity that is exactly vertical. The angle is not a field. Drag is not subtracted. In air the real peak is often lower, and the time going up and the time coming down stop being equal. Mass does not enter the formula: in the balance ½ v₀² = g H_max, mass appears on both sides and drops out. A throw with an angle is on the projectile-motion calculator. The peak at an angle is on a separate page. A throw off an edge with a horizontal speed is on the horizontal-throw calculator. Distance from time, starting from rest, is on the h = ½ g t² calculator. Time from a height, with no initial speed, is on the t = √(2h/g) calculator.
The unit switch in the header changes the speed to ft/s and g to ft/s², and it converts numbers already entered. Time stays in seconds. The sentences on the page stay in SI units as the reference. The chart axis shows height in the unit selected in the header.
The examples use the same formula. 15 m/s upward gives 11.47 m. 10 m/s and t = 1 s give v = 0.19 m/s and h = 5.10 m. 5 m/s downward after 1.5 s gives a drop of 18.54 m and a speed of 19.72 m/s. 15 m/s at g = 1.62 m/s² gives 69.44 m. The button “Fill this example” enters those numbers and refreshes the panel and the chart. The fields start empty.
Formula
v(t) = v0 − g t h(t) = v0t − ½ g t2
Hmax = v02 / (2g) tup = v0 / g T = 2 v0 / g
Downward throw: v(t) = v0 + g t, drop distance h(t) = v0t + ½ g t2. A climb peak is not computed in that mode.
Model: uniformly accelerated motion on a vertical line (Galileo, Discorsi, 1638; school form H_max = v₀²/(2g)). The angle is not an input. Upward order of operations: square the speed, divide by 2g, then take the time v₀/g and the doubled flight time. When a time is entered, the calculator substitutes it into v(t) and h(t).
Theory, derivation, and domain
Definition of a vertical flight
A vertical throw is motion in which the initial velocity is parallel to the vertical. In the upward mode that velocity points against the weight. In the downward mode it points with the weight. The height h is measured from the launch level. There is no horizontal axis and no range Z.
Where H_max comes from
The condition v(t) = 0 gives t_up = v₀/g. Substituting that time into h(t) gives H_max = v₀²/(2g). The same result follows from an energy balance with no drag: ½ v₀² = g H_max. The calculator computes the quotient, not an energy in joules. There is no mass field. For 15 m/s and g = 9.81 m/s² the square of the speed is 225 (m/s)² and 2g is 19.62 m/s². The panel shows 11.47 m.
Time going up and time coming down
With no drag, the time from the peak back to launch level equals the climb time. For 15 m/s both parts are 1.53 s, and the whole flight is 3.06 s. The panel does not build the height from the rounded time 1.53 s. The product 15 × 1.53 / 2, rounded to two decimal places, is 11.48 m, and the calculator shows 11.47 m. Substituting the rounded flight time 3.06 s into h(t) gives −0.03 m, not zero. Maximum height and time are computed from the full quotients and only then rounded to two decimal places.
Why the height grows with the square of the speed
Doubling v₀ multiplies the height by four before rounding and doubles the time. A change from 15 m/s to 30 m/s raises the height from 11.47 m to 45.87 m, and the climb from 1.53 s to 3.06 s. A change from 10 m/s to 20 m/s raises the height from 5.10 m to 20.39 m. Four times 5.10 m is 20.40 m. The one-centimeter difference on the panel comes from rounding, not from a different formula.
Downward throw
The signs change when the initial velocity points with the weight: v(t) = v₀ + g t, and the drop distance is h(t) = v₀ t + ½ g t². A climb peak is not computed. For 5 m/s and t = 1.5 s at g = 9.81 m/s² the speed is 19.72 m/s (71.0 km/h) and the drop is 18.54 m. At v₀ = 0 and t = 1 s this mode gives a distance of 4.91 m, which is ½ g t². The upward mode at v₀ = 0 and t = 1 s gives a height of −4.91 m, because the axis points up, and the object is below the launch level. A drop from rest, with time as the input and a positive distance, is on the h = ½ g t² calculator.
What this formula does not compute
There is no angle, mass, drag, or force. An empty g field means exactly 9.81 m/s². Standard gravity, 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 11.47 m and 1.53 s. When the landing is not at the launch level, T = 2 v₀/g does not describe the impact. When the entered time exceeds T, the formula still reports the state: for 15 m/s and t = 4 s the speed is −24.24 m/s and the height is −18.48 m, below the launch level. The calculator does not stop the motion at h = 0.
How to use
- Choose the direction: upward or downward. Upward, the calculator computes the peak and the times. Downward, it computes the speed and the drop distance when you enter a time.
- Enter the initial speed in meters per second, for example 15. That is the speed at launch, not a mean over the climb.
- Leave t empty unless you want v(t) and h(t) at a chosen instant. Leave g empty for 9.81 m/s², or enter another value, for example 1.62.
- Click Calculate or wait for the live result. For 15 m/s upward and an empty g field, the calculator shows a height of 11.47 m and the times 1.53 s / 3.06 s.
- Check the panel: the initial speed, g, and the speed on return. Clearing the form hides the panel. An angle and a horizontal range belong on another page.
Symbol glossary
v₀ is the initial vertical speed, H_max is the peak height, t_up is the climb time, and T is the flight back to launch level. For 15 m/s at g = 9.81 m/s² the height is 11.47 m.
- v₀
- Initial speed [m/s]. At 15 m/s the height is 11.47 m. At 30 m/s the height is 45.87 m. A speed of zero upward gives a height of 0.00 m and times of 0.00 s / 0.00 s.
- g
- Acceleration [m/s²]. An empty field means 9.81, not 9.80665. For 15 m/s both values show 11.47 m on the panel.
- t
- Optional time [s]. An empty field leaves v(t) and h(t) without a number. For 10 m/s and t = 1 s the speed is 0.19 m/s and the height is 5.10 m.
- H_max
- Peak height on an upward flight [m], H_max = v₀²/(2g). In the downward mode the panel shows “not applicable”.
- t_up
- Climb time [s], t_up = v₀/g. For 15 m/s it is 1.53 s. It is not a separate field.
- T
- Flight time back to launch level [s], T = 2 v₀/g. For 15 m/s it is 3.06 s.
- v(t)
- Speed at time t. Upward: v₀ − g t. Downward: v₀ + g t. For 5 m/s downward and t = 1.5 s it is 19.72 m/s (71.0 km/h).
- h(t)
- Height from the launch level. Upward: v₀ t − ½ g t². A negative value means a point below the launch. For 15 m/s and t = 4 s it is −18.48 m.
Real-life examples
Example 2
The same upward axis, at 10 m/s, at the instant t = 1 s. The climb lasts 1.02 s, so this instant is still before the peak. The maximum height and the height at 1 s both display as 5.10 m.
H_max = 5.10 m
Example 3
A speed of 20 m/s upward, with no time. That is twice 10 m/s, and the height is close to four times 5.10 m. The panel shows 20.39 m.
H_max = 20.39 m
Example 4
A short upward throw at 8 m/s. Air resistance is not subtracted. The time to the peak is 0.82 s.
H_max = 3.26 m
Example 5
A stream of water starts straight up at 12 m/s. The result is the vacuum peak at constant g. The flight back to launch level takes 2.45 s.
H_max = 7.34 m
Example 6
A center of mass is given a speed of 4 m/s upward. The calculator reports the vacuum peak of that speed, not a measured jump height.
H_max = 0.82 m
Example 7
A downward throw at 5 m/s, read at 1.5 s. The height field shows “not applicable”. The third field reports a speed of 19.72 m/s and the drop distance.
drop = 18.54 m
Example 8
The same 15 m/s upward, 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.
H_max = 69.44 m
Frequently asked questions
What does this calculator compute, and from which formula?
The calculator finds the maximum height of an upward vertical throw from H_max = v₀²/(2g) and the climb time from t_up = v₀/g. The flight back to launch level is 2 v₀/g. For 15 m/s at g = 9.81 m/s² the panel shows a height of 11.47 m, a climb of 1.53 s, and a flight of 3.06 s. The height is computed from the full quotient, not from the product 15 × 1.53 / 2.
Which units should I enter for speed, time, and g?
Enter the speed in meters per second, the time in seconds, and g in meters per second squared. The height is in meters. After you switch the header to US units, the speed is in ft/s 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 15 m/s both values show the same height of 11.47 m and the same time of 1.53 s 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 doubling the speed not double the height?
Because the formula squares the speed. Doubling v₀ multiplies the height by four before rounding and doubles the time. A change from 15 m/s to 30 m/s raises the height from 11.47 m to 45.87 m, and the climb from 1.53 s to 3.06 s. A change from 10 m/s to 20 m/s raises the height from 5.10 m to 20.39 m.
How is an upward throw different from a downward throw and from free fall?
Upward, the axis points against the weight, so the calculator computes a peak and the times. Downward, the initial velocity points with the weight, a peak is not computed, and the formulas use a plus sign. Free fall starts at v₀ = 0 and computes a distance from a time. A horizontal throw has a range Z. Projectile motion accepts an angle. This page has no horizontal component.
What does a speed of zero mean?
Upward, with no time, the height is 0.00 m and both times are 0.00 s. Upward at v₀ = 0 and t = 1 s the height is −4.91 m, because the axis points up. Downward at v₀ = 0 and t = 1 s the drop is 4.91 m. A negative number in the speed field is rejected. An empty speed field leaves 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 lower the peak and does not split the upward time from the downward time. In air the real peak is often lower, and those two times stop being equal. There is no mass field.
What does the panel show when I enter a time t?
Upward, the third field reports v(t) = v₀ − g t and h(t) = v₀ t − ½ g t². For 10 m/s and t = 1 s the speed is 0.19 m/s (0.7 km/h) and the height is 5.10 m. Downward, the same field reports v = v₀ + g t and the drop distance. For 5 m/s and t = 1.5 s the speed is 19.72 m/s and the drop is 18.54 m. Without an entered time, the panel does not show v(t) or h(t).
What happens when the time is longer than the return to launch level?
The calculator still substitutes the time into the formula and does not stop the motion at height zero. For 15 m/s the flight time is 3.06 s. At t = 4 s the speed is −24.24 m/s and the height is −18.48 m, below the launch level. The formula T = 2 v₀/g describes the return to launch level, not an impact on lower ground.
What do the panel and the chart show?
The panel reports the maximum height, the time to the peak and the flight time, the state v(t) and h(t), the initial speed, the g in use, and the speed on return. For 15 m/s the speed on return is 15.00 m/s. The chart, only for an upward throw with v₀ greater than zero, draws the height at six instants of the flight. 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 15 m/s upward the height is then 69.44 m and the climb time is 9.26 s. At the Earth value 9.81 m/s² the same 15 m/s give 11.47 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. 12.5 and 12,5 are the same speed. For that value, upward and at g = 9.81 m/s², the height is 7.96 m and the climb time is 1.27 s.
How do I calculate maximum height with H_max = v₀²/(2g)?
Divide the square of the initial speed by twice g. At 15 m/s and g = 9.81 m/s²: 225/(2·9.81) ≈ 11.47 m.
What is the climb time at 15 m/s?
t↑ = v₀/g = 15/9.81 ≈ 1.53 s. Time back to launch level with no drag is about 3.06 s.
Why does doubling v₀ quadruple H_max?
The formula has v₀². At 30 m/s with the same g the height is about 45.87 m, four times the height at 15 m/s.
How is a vertical throw different from free fall?
Here v₀ points up or down. Free fall is the v₀ = 0 case and has a separate page with t = √(2h/g).
How is this page different from oblique projectile motion?
Motion is only on the vertical axis; there is no horizontal component and no range. Angle and range belong to the oblique-throw calculators.
Knowledge sources
The formula, the account of accelerated motion, and the units come from mechanics textbooks and from metrology sources. The page does not invent publication identifiers.
- Wikipedia — Projectile motion
- Wikipedia — Two New Sciences (Galileo, Discorsi, 1638)
- Wikipedia — Standard gravity (9.80665 m/s²)
- BIPM — The International System of Units (SI Brochure)
- NIST SP 330 — SI units guide
- NASA — Moon Fact Sheet
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics, vol. 1, chapter on projectile motion.
- Young, H. D.; Freedman, R. A., University Physics, chapter on motion in two dimensions.
Page updated November 3, 2026.