Mechanical Energy Calculator

What total mechanical energy adds up

Mechanical energy is the energy a moving or elevated object carries because of its motion and its position, and it comes in two parts. Kinetic energy is the motion part. Gravitational potential energy is the position part. Add them and you have total mechanical energy, written ME = KE + PE.

Each part has a short formula. Kinetic energy is KE = 0.5 x m x v^2, where m is mass in kilograms and v is speed in meters per second. Gravitational potential energy is PE = m x g x h, where g is 9.81 m/s^2 near the surface of the Earth and h is height in meters. Put together: ME = 0.5 x m x v^2 + m x g x h.

Both parts come out in joules, the SI unit of energy, so they add cleanly. A 2 kg object moving at 3 m/s and sitting 5 m above your chosen zero has 9 J of kinetic energy and 98.1 J of potential energy, for a total of 107.1 J. The calculator below does that sum for any mass, speed and height you enter, and it can add a spring term if the object is stretching or compressing one.

Kinetic energy: speed counts twice

The v^2 in the kinetic energy formula is the detail that changes every estimate. Double the speed and the energy does not double; it quadruples. A 2 kg object at 3 m/s carries 0.5 x 2 x 3 x 3 = 9 J. The same object at 6 m/s carries 0.5 x 2 x 6 x 6 = 36 J, four times as much, from a speed that only went up by 3 m/s.

Mass, by contrast, counts once. Double the mass at the same speed and you double the energy. That asymmetry is why speed limits matter more than vehicle weight in crash energy, and why a pitched baseball carries a surprising amount of energy for its 145 grams: 0.145 kg at 40 m/s holds 0.5 x 0.145 x 1,600 = 116 J.

Gravitational potential energy and the number g

Potential energy near the ground is PE = m x g x h. The g is the acceleration of gravity, and this calculator fixes it at 9.81 m/s^2. The exact standard value, used for unit conversions, is 9.80665 m/s^2 (see the NIST Guide to the SI, appendix B), and 9.81 is that value rounded. For anything happening in a room, a yard or a stadium, the rounding never shows up in the second decimal.

One useful property of g: it is a constant of nature, not a rate or a tax. There is no yearly update to look up. Real gravity does vary slightly with latitude, about 9.78 m/s^2 at the equator and 9.83 at the poles, and a bit less on a mountain, but those shifts are under half a percent and outside anything a homework problem or a hobby project needs.

Height h is measured from a zero point you choose, and the choice is yours to make: the floor, the desk, the ground below a bridge. That is why potential energy on its own can look arbitrary. What has physical meaning is the change in potential energy, and moving the zero point moves every height by the same amount, so every difference stays put. The HyperPhysics page on gravitational potential energy walks through this choice in more detail.

The spring term, when you need it

A third kind of mechanical energy shows up whenever the object stretches or squeezes a spring. Elastic potential energy is 0.5 x k x x^2, where k is the spring constant in newtons per meter and x is how far the spring is stretched or compressed from its resting length, in meters. A spring with k = 200 N/m pulled 0.1 m past rest stores 0.5 x 200 x 0.01 = 1 J.

With the spring term included, the full sum is ME = 0.5 x m x v^2 + m x g x h + 0.5 x k x x^2. In the worked example later in this page, that 1 J lifts the total from 107.1 J to 108.1 J.

Leave the spring boxes at zero and the term contributes nothing, which is the right answer for most problems: a thrown ball, a book on a shelf and a car on a highway involve no spring. Note that x must be the stretch distance in meters, not centimeters. A 10 cm stretch is 0.1 m, and entering 10 would report 100 times too much energy, because the x is squared.

A 2 kg object at 3 m/s and 5 m up, step by step

Here is the example the calculator opens with, one term at a time. The object has a mass of 2 kg, a speed of 3 m/s and a height of 5 m above the floor.

Kinetic energy first: KE = 0.5 x m x v^2 = 0.5 x 2 x 3^2 = 0.5 x 2 x 9 = 9 J. Potential energy next: PE = m x g x h = 2 x 9.81 x 5 = 98.1 J. Total: ME = 9 + 98.1 = 107.1 J.

Now add the spring. Attach a spring of k = 200 N/m and stretch it 0.1 m: the spring term is 0.5 x 200 x 0.01 = 1 J, and the total becomes 9 + 98.1 + 1 = 108.1 J. Type those five numbers into the calculator and it returns exactly those four results, which is the fastest way to check your own hand arithmetic.

Notice the balance between the terms. The object is crawling along at walking speed, yet its potential energy is eleven times its kinetic energy, because 5 m of height is worth a lot. Lower the object to 1 m and the potential part drops to 19.6 J, and the kinetic part would dominate.

Everyday objects, side by side

Numbers mean more in company. The table below runs the same formula over a few familiar objects, using g = 9.81 m/s^2 and rounding to sensible precision. Heights are measured from the ground, speeds in meters per second (25 m/s is about 56 mph, 40 m/s is about 89 mph).

A few reads off that table. The book's 14.7 J comes from 1 x 9.81 x 1.5, and it is the energy that goes into the thud when the book falls. The person climbing 10 m of stairs banks 70 x 9.81 x 10 = 6,867 J of potential energy, which the body pays for in food Calories, and pays at a poor exchange rate. The highway car shows why mass once more: at only 25 m/s it already holds over 468,000 J, roughly 4,000 times the baseball, because it is ten thousand times as heavy and moving fast.

Where the energy goes: trading height for speed

Total mechanical energy earns its keep in problems where the two parts trade. Drop the 2 kg object from 5 m, starting at rest. At the top it holds 98.1 J, all potential. Ignoring air resistance, all 98.1 J arrive at the floor as kinetic energy, and the speed that carries it comes from setting 0.5 x m x v^2 = m x g x h. The mass cancels and v = sqrt(2 x g x h) = sqrt(2 x 9.81 x 5) = 9.9 m/s.

That cancellation is worth remembering: how fast something lands after a fall does not depend on how heavy it is, only on how far it fell. A 2 kg object and a 20 kg one dropped from 5 m both hit at 9.9 m/s. The heavier one carries ten times the energy when it arrives, which is why the landing sounds different.

The same trade runs uphill in reverse. The roller coaster car in the table holds 247,150 J at the top of its 30 m lift hill if it crawls, but at 20 m/s at the top it holds 100,000 J of that as speed. Designers spend the 147,150 J of height buying speed on the way down, and the total stays close to constant until friction and drag tax it. That tax is real: a coaster loses maybe a third of its mechanical energy over a full circuit to brakes and wind, which is why the second hill is always shorter than the first.

Limits, units and the checks worth making

The formula assumes no friction or drag. Real objects shed mechanical energy to air resistance and rubbing surfaces, and that energy turns into heat rather than disappearing. A feather dropped from 5 m never comes close to 9.9 m/s. For dense objects falling a few meters, the simple sum is close; for light or fast cases, treat it as a ceiling.

Rotation is not included. A rolling ball, a spinning wheel and an orbiting planet carry rotational kinetic energy of 0.5 x I x w^2 on top of the straight line part, where I is the moment of inertia. This page covers straight line motion and springs, which is what most classroom and household questions ask.

Units in, units out. Keep mass in kilograms, speed in meters per second and height in meters, and the answer lands in joules with no conversion factor. A pound is 0.4536 kg, a mile per hour is 0.447 m/s, and a foot is 0.3048 m; convert those first. The joule is kg x m^2 / s^2 in SI base units, per the NIST Guide to the SI.

Sanity checks. Kinetic energy can never be negative, so a negative answer means a negative entered where it should not be. Potential energy can read negative if your zero point is above the object, which is legal, but say so in your work. And if a result looks enormous, check that speed was in meters per second: entering 60 for mph instead of 26.8 inflates the kinetic term by a factor of five.