Mechanical Energy Calculator

Free Online Mechanical Energy Calculator (ME = KE + PE)

This mechanical energy calculator instantly solves for total mechanical energy, mass, velocity, or height using the fundamental formula ME = KE + PE = 1/2 × m × v² + m × g × h — mechanical energy equals kinetic energy plus gravitational potential energy. Enter the mass, velocity, and height of any object and the calculator converts everything automatically, works through a complete step-by-step solution, and draws a labeled diagram of the moving, elevated object alongside a KE/PE/ME bar chart with every value printed directly on the bars, so the entire energy breakdown is visible at a glance, not just a single final number.

Whether you're a physics student solving a mechanical energy word problem, checking a roller-coaster or pendulum energy calculation, verifying conservation of energy in a falling-object problem, or just curious how much total energy a moving, elevated object carries, this calculator handles kilograms, grams, pounds, ounces, and metric tons for mass; meters/second, kilometers/hour, miles/hour, feet/second, and knots for velocity; meters, centimeters, kilometers, feet, inches, and miles for height; and joules, kilojoules, calories, kilocalories, foot-pounds, and ergs for energy — plus gravity presets for Earth, the Moon, Mars, and Jupiter, or a custom value. It always shows its complete working, step by step.

What Is Mechanical Energy? (Definition and Formula)

Mechanical energy is the total energy an object has due to its motion and its position — it is simply the sum of two energy types you may already know separately: kinetic energy (the energy of motion) and gravitational potential energy (the energy of height or position). Any object that is both moving and elevated above a reference point — a roller-coaster car cresting a hill, a thrown ball in mid-flight, or a pendulum swinging through its arc — carries mechanical energy made up of both pieces at once.

The formula for total mechanical energy combines the two familiar formulas into one:

• Mechanical Energy (ME) = Kinetic Energy (KE) + Potential Energy (PE)
• ME = 1/2 × mass (m) × velocity (v)^2 + mass (m) × gravity (g) × height (h)
• The SI unit of mechanical energy is the joule (J), same as kinetic energy, potential energy, and work.
• Example: a 500 kg roller-coaster car moving at 12 m/s at a height of 20 m has ME = (0.5 × 500 × 12²) + (500 × 9.81 × 20) = 36,000 + 98,100 = 134,100 J

Kinetic Energy and Potential Energy — The Two Building Blocks

To understand mechanical energy, it helps to see its two components side by side. Kinetic energy (KE = 1/2mv²) depends on mass and the square of velocity — it exists only while the object is moving, and grows very quickly with speed since velocity is squared. Potential energy (PE = mgh) depends on mass, gravity, and height — it exists purely because of position above a reference level, whether or not the object is moving at all. Mechanical energy simply adds these two together, which is exactly why the bar chart on this page shows KE and PE as separate bars before showing ME as their combined total — you can see precisely how much of the total energy comes from motion versus how much comes from height.

How to Find Mass from Mechanical Energy, Velocity, and Height

If you already know an object's total mechanical energy, its velocity, and its height, you can rearrange the formula to solve for mass — because both KE and PE scale directly with mass, it factors out of the entire equation.

• Mass (m) = Mechanical Energy (ME) ÷ (1/2 × v² + g × h)
• Example: ME = 96,300 J at v = 12 m/s and h = 20 m on Earth → m = 96,300 ÷ (0.5 × 144 + 9.81 × 20) = 96,300 ÷ (72 + 196.2) = 96,300 ÷ 268.2 ≈ 359.1 kg

How to Find Velocity or Height from Mechanical Energy

If mass and height are known along with total mechanical energy, first subtract the potential energy (mgh) from ME to find the kinetic energy, then rearrange KE = 1/2mv² for velocity. Symmetrically, if mass and velocity are known, first subtract the kinetic energy (1/2mv²) from ME to find the potential energy, then rearrange PE = mgh for height.

• Velocity (v) = √( 2 × (ME − m×g×h) ÷ m )
• Height (h) = (ME − 1/2×m×v²) ÷ (m×g)
• If the subtraction produces a negative number — meaning the other energy term alone already exceeds the total ME — there is no real solution, since kinetic energy and height can never be negative.

How to Use This Mechanical Energy Calculator — Step by Step

Choose what you want to find — Mechanical Energy, Mass, Velocity, or Height — from the dropdown. Then fill in the remaining known values in any supported unit, and select which gravity to use — Earth by default, or the Moon, Mars, Jupiter, or a custom value for off-world problems. The calculator instantly converts everything to a common base unit, applies ME = KE + PE (or its rearranged form), and displays the result in your chosen unit alongside the separate KE and PE contributions.

Read the diagram above the step-by-step solution: the object is drawn at its height with its velocity vector, the amber box shows total mechanical energy using your real numbers, and the bar chart beneath it plots KE, PE, and ME side by side with the exact value labeled on every single bar — so you can see the full energy breakdown at a glance, not just the final total.

Worked Example — Finding Mechanical Energy

A 500 kg roller-coaster car is moving at 12 m/s while 20 m above the ground on Earth (g = 9.81 m/s²). First find kinetic energy: KE = 0.5 × 500 × 12² = 0.5 × 500 × 144 = 36,000 J. Next find potential energy: PE = 500 × 9.81 × 20 = 98,100 J. Finally add them: ME = 36,000 + 98,100 = 134,100 J. This total tells you the car's combined energy of motion and position at that exact instant on the track.

Worked Example — Finding Mass from Mechanical Energy

Suppose an object has 96,300 J of mechanical energy while moving at 12 m/s at a height of 20 m on Earth. Using m = ME ÷ (1/2v² + gh): the per-kilogram energy term is 0.5 × 12² + 9.81 × 20 = 72 + 196.2 = 268.2 J/kg. Dividing: m = 96,300 ÷ 268.2 ≈ 359.1 kg.

Understanding the Diagram and Bar Chart: Reading the Values Directly

The top of the diagram shows the object itself, positioned at its actual height above the ground with a dashed guide line labeled h, and a velocity arrow labeled v — both using your real entered or calculated numbers. Just below, an amber box states the total mechanical energy plainly. The bar chart underneath is the key to understanding the split: three bars — KE, PE, and ME — are drawn to scale against each other, and the exact joule value (or your chosen energy unit) is printed directly above every bar, with the small formula label (½mv², mgh, or KE + PE) underneath each one. A caption above the chart also spells out the addition, such as '36,000 J + 98,100 J = 134,100 J', so the relationship between the three numbers is impossible to miss without needing to do any mental math yourself.

Conservation of Mechanical Energy

One of the most important ideas connected to this formula is the conservation of mechanical energy: in the absence of air resistance and friction, an object's total mechanical energy stays constant even as it trades kinetic energy for potential energy and back again. A ball thrown upward loses velocity (and kinetic energy) as it rises but gains height (and potential energy) at exactly the same rate, so ME stays the same throughout the flight — right up until it starts falling again and the process reverses. This principle is exactly why a pendulum, a roller coaster, or a bouncing ball (ignoring losses) returns to the same height it started from: the mechanical energy hasn't gone anywhere, it has simply shifted between the KE and PE bars in the chart above.

Real-World Applications of Mechanical Energy

The mechanical energy formula and the conservation principle behind it appear throughout engineering, sports, and everyday physics:

• Roller coasters and amusement rides — designers calculate the mechanical energy at every point of the track to ensure cars have enough speed to complete loops and hills safely.
• Pendulum clocks and swings — the smooth back-and-forth conversion between kinetic and potential energy keeps a pendulum's total mechanical energy constant (ignoring friction).
• Hydroelectric and wave power — falling or moving water carries mechanical energy that turbines convert into electricity.
• Vehicle safety engineering — total mechanical energy at the moment of impact helps engineers design crumple zones and restraint systems.
• Sports science — analyzing a diver, gymnast, or javelin's combined energy of motion and height helps coaches understand performance and technique.

Tips for Getting Accurate Results

Always convert mass to kilograms, velocity to meters per second, and height to meters before manual calculation — this calculator does that automatically, but understanding the base units helps you sanity-check the answer. Double-check which gravity value applies to your problem; using Earth's 9.81 m/s² by default when a problem specifies the Moon or another planet is one of the most common mistakes. When solving for velocity or height, remember that the subtraction step (ME − PE or ME − KE) must stay positive — a negative result means the given mechanical energy is too small for the other stated value, so no real solution exists, and the calculator will flag this rather than return a misleading number. Finally, use the bar chart as a visual check: the KE and PE bars should always add up exactly to the ME bar.