Height and Velocity Calculator

Example data table

Metric examples using v² = u² + 2·g·h (downward).

Formula used

This calculator uses constant-acceleration kinematics for vertical motion:

• v² = u² + 2·a·h (links height and velocities)
• v = u + a·t (links time and velocity)
• h = u·t + ½·a·t² (links time and displacement)

Here, a equals +g for downward motion and −g for upward motion.

How to use this calculator

1. Select a Solve option to choose what you want to find.
2. Pick a Unit system and the Motion direction.
3. Enter the required inputs (leave unused fields empty).
4. Optionally set gravity and decimal places for rounding.
5. Press Calculate to view results above the form.
6. Use Download CSV or Download PDF to export.

Height and velocity guide

1) What this calculator solves

This tool links vertical height, time, and speed using constant-acceleration motion. Choose a solve option to compute final velocity, initial velocity, height (displacement), or time. It is useful for drops, throws, elevator tests, and lab kinematics where acceleration is approximately constant.

2) Units and sign convention

You can work in metric (m, m/s) or imperial (ft, ft/s). Direction matters: downward motion typically uses +g and upward motion uses −g. If your velocity sign does not match your direction choice, you may see a negative result that simply indicates the opposite direction.

3) Gravity values you can enter

Standard gravity is about 9.80665 m/s², which is 32.174 ft/s². You can adjust g for local conditions or for experiments (for example, 9.81 is a common classroom value). Changing g affects every result, especially time estimates.

4) Height to speed (quick data points)

Ignoring air resistance and starting from rest, speed grows with the square root of height. A 10 m drop gives about 14.0 m/s (≈ 31 mph), while a 100 m drop gives about 44.3 m/s (≈ 99 mph). In feet, a 100 ft drop gives about 80.3 ft/s.

5) Speed to height (reverse lookup)

If you know the takeoff speed for an upward toss, the peak rise (above the launch point) is h = u²/(2g). For example, an upward launch of 20 m/s rises about 20.4 m. A launch of 30 ft/s rises about 14.0 ft.

6) Time-based results

With constant acceleration, velocity changes linearly with time: v = u + at. Starting from rest in free fall, t = 1 s gives about 9.81 m/s, and t = 5 s gives about 49.0 m/s. Use the time option when you have a stopwatch measurement.

7) Energy viewpoint for sanity checks

A helpful check is the energy form v² − u² = 2gh. It shows that doubling height does not double speed; speed increases by √2. If your computed speed seems too large, verify that units are consistent and that height is not entered in centimeters or inches by mistake.

8) Practical limits and accuracy

Real objects experience air drag, so measured speeds are often lower than ideal predictions, especially for long drops. Wind, launch angle, and sensor delays also add error. For best accuracy, use short intervals, measure height carefully, and keep units consistent across every input field.

FAQs

1) Is air resistance included?

No. The calculator assumes constant acceleration and ignores drag. For long falls or light objects, real speeds can be significantly lower than the ideal result.

2) Why did I get a negative height or velocity?

A negative sign usually means the motion is opposite to the selected direction. Recheck your direction choice and whether you entered upward velocities as positive or negative.

3) What gravity value should I use?

Use 9.80665 m/s² (or 32.174 ft/s²) for standard calculations. For quick work, 9.81 m/s² is fine. Change g only if you have a measured local value.

4) Can I calculate the peak height of an upward throw?

Yes. Select the option that solves for height and set final velocity to zero at the top. The output gives the rise above the launch point, assuming constant acceleration.

5) Which formula is best when time is unknown?

Use the relation v² − u² = 2ah. It eliminates time and is often the most direct way to link height and velocities when you do not know t.

6) Why does the time solution sometimes show “no real solution”?

Time comes from a quadratic equation. If the discriminant is negative, the chosen inputs cannot occur under constant acceleration (for example, too little height for the requested speed change).

7) Does the calculator handle mixed units?

No. Pick one unit system and keep all inputs in that system. Mixing meters with ft/s (or vice versa) will produce incorrect results.