Mechanical energy is the sum of the kinetic energy and the potential energy of a body: ME = KE + PE. It is the energy an object has because of its motion and because of its position or configuration, and its SI unit is the joule (J), where 1 J = 1 N·m = 1 kg·m²/s². A 2 kg stone held 20 m above the ground has 400 J of mechanical energy, all of it potential; the instant before it lands it still has 400 J, now all kinetic. That constancy is the point of the idea, and it holds whenever only conservative forces act on the body.
What mechanical energy means
Energy in mechanics comes in two macroscopic forms, and mechanical energy is just their total.
- Kinetic energy (KE) is the energy of motion. Anything with mass that is moving has it: a rolling ball, a spinning flywheel, a falling raindrop.
- Potential energy (PE) is stored energy that a body has because of where it is or how it is arranged. A book on a shelf, a compressed spring and a drawn bow all hold it.
Add them and you get mechanical energy. Notice what is excluded: the random jiggling of molecules inside the body is thermal energy, not mechanical energy, even though it is also kinetic at the microscopic level. Mechanical energy counts only the organised, whole-body motion and position.
Kinetic energy: KE = ½mv²
The formula falls out of the definition of work. Push a body of mass m from rest with a constant force F, and it accelerates at a = F/m over a distance s. From v² = u² + 2as with u = 0, s = v²/2a. The work you did is:
W = F × s = ma × v²/(2a) = ½mv²
That work is now stored in the body as kinetic energy, so KE = ½mv². Two features matter in problems. KE is always positive, because v is squared and mass cannot be negative. And it goes with the square of speed, so doubling the speed quadruples the kinetic energy. A car at 60 km/h carries four times the kinetic energy it carries at 30 km/h, which is why braking distance and crash severity climb so steeply with speed.
A body that spins as well as moves also carries rotational kinetic energy, ½Iω², where I is the moment of inertia and ω the angular speed. A rolling wheel has both, and its total kinetic energy is ½mv² + ½Iω².
Potential energy: gravitational and elastic
Gravitational potential energy, PE = mgh
Lift a body of mass m through a height h at steady speed. You must apply an upward force equal to its weight, mg, over a distance h, so the work done is:
W = F × h = mgh, and this is stored as PE = mgh
Here g is the acceleration due to gravity, 9.8 m/s² on Earth, often rounded to 10 m/s² in classroom work. The height h is measured from whatever reference level you choose. That choice is free, because only changes in potential energy have physical meaning; pick the ground, the table top or the floor of the lab, then stick to it for the whole problem.
Elastic potential energy, PE = ½kx²
For a spring obeying Hooke’s law, the restoring force is F = kx, so it rises from 0 to kx as you stretch or compress it by x. The average force over that stretch is ½kx, and the work done is:
W = average force × distance = ½kx × x = ½kx²
So PE = ½kx², where k is the spring constant in N/m and x is the deformation from the natural length, in metres. A spring of k = 200 N/m compressed by 10 cm stores ½ × 200 × (0.1)² = 1 J.
Examples of mechanical energy with numbers
Taking g = 10 m/s² and measuring heights from the ground:
| Object | Kinetic energy | Potential energy | Total mechanical energy |
|---|---|---|---|
| 1 kg book resting on a 2 m shelf | 0 J (at rest) | 1 × 10 × 2 = 20 J | 20 J |
| 1000 kg car at 20 m/s on a level road | ½ × 1000 × 20² = 200 kJ | 0 J (at the reference level) | 200 kJ |
| 1 kg hammer head at 8 m/s just before it strikes | ½ × 1 × 8² = 32 J | 0 J | 32 J |
| 1 kg of water at the top of a 50 m dam | 0 J | 1 × 10 × 50 = 500 J | 500 J |
| Spring, k = 200 N/m, compressed 0.1 m | 0 J | ½ × 200 × 0.1² = 1 J | 1 J |
| 0.15 kg cricket ball at 30 m/s, 2 m above the pitch | ½ × 0.15 × 30² = 67.5 J | 0.15 × 10 × 2 = 3 J | 70.5 J |
The cricket ball row is the useful one: most real objects carry both forms at once, and mechanical energy is simply their sum at that instant.
Conservation of mechanical energy
If only conservative forces do work on a body, its total mechanical energy stays constant: KE + PE = constant, or KE_1 + PE_1 = KE_2 + PE_2 between any two points of the motion.
A conservative force is one for which the work done depends only on the start and end positions, not on the path taken, and for which the work around any closed loop is zero. Gravity, the spring force and electrostatic force are conservative. Friction, air drag, viscous resistance and the forces in a collision that deforms metal are not.
So the conditions are strict, and they are the part students skip:
- No friction between the body and any surface.
- No air resistance or fluid drag.
- No external agent adding energy, such as an engine, a motor or a hand still pushing.
- No energy leaving as sound, heat or permanent deformation.
Worked example: a falling body
Drop a 2 kg stone from rest at 20 m, with g = 10 m/s² and no air resistance. Total mechanical energy at the start = mgh = 2 × 10 × 20 = 400 J. Track it down.
| Height | Distance fallen | v² = 2gs | KE = ½mv² | PE = mgh | Total |
|---|---|---|---|---|---|
| 20 m | 0 m | 0 | 0 J | 2 × 10 × 20 = 400 J | 400 J |
| 15 m | 5 m | 2 × 10 × 5 = 100 | ½ × 2 × 100 = 100 J | 2 × 10 × 15 = 300 J | 400 J |
| 10 m | 10 m | 2 × 10 × 10 = 200 | ½ × 2 × 200 = 200 J | 2 × 10 × 10 = 200 J | 400 J |
| 0 m | 20 m | 2 × 10 × 20 = 400 | ½ × 2 × 400 = 400 J | 0 J | 400 J |
Potential energy converts into kinetic energy joule for joule. The landing speed is v = √400 = 20 m/s, which you can get straight from mgh = ½mv², giving v = √(2gh) with the mass cancelling. That is why a heavy stone and a light one, dropped together in vacuum, land together.
Worked example: a simple pendulum
A bob of 0.5 kg hangs on a 1 m string and is pulled aside to 60° from the vertical. Its rise above the lowest point is h = L(1 − cos 60°) = 1 × (1 − 0.5) = 0.5 m. At release, KE = 0 and PE = 0.5 × 10 × 0.5 = 2.5 J. At the bottom of the swing all of it is kinetic, so ½ × 0.5 × v² = 2.5, giving v² = 10 and v = 3.16 m/s. It then climbs to the same height on the far side. The string tension does no work at all, because it always acts at right angles to the motion.
Worked example: a spring-mass system
A 0.2 kg block on a frictionless table is pressed against a spring of k = 200 N/m, compressing it 0.1 m. Stored elastic PE = ½ × 200 × 0.01 = 1 J. Released, the block leaves the spring with 1 J of kinetic energy: ½ × 0.2 × v² = 1, so v² = 10 and v = 3.16 m/s. In an oscillating spring-mass system the energy shuttles between elastic PE at the extremes and KE at the centre, and the total stays at 1 J.
Worked example: a roller coaster
A 500 kg car starts at rest at the top of a 30 m hill on a frictionless track. Total ME = 500 × 10 × 30 = 150,000 J, fixed for the whole ride. At a point 10 m above the ground, PE = 500 × 10 × 10 = 50,000 J, so KE = 150,000 − 50,000 = 100,000 J and ½ × 500 × v² = 100,000 gives v = 20 m/s. Two consequences follow directly: no later hill can be taller than 30 m, and the shape of the track between the two points makes no difference to the speed, only the height does.
What happens when friction acts
Take the same 2 kg stone, but slide it down a 20 m ramp instead of dropping it, and measure its speed at the bottom as 15 m/s rather than 20 m/s.
- Mechanical energy at the top: 2 × 10 × 20 = 400 J
- Mechanical energy at the bottom: ½ × 2 × 15² = 225 J
- Mechanical energy missing: 400 − 225 = 175 J
Those 175 J are not destroyed. Friction converted them into thermal energy in the ramp and the stone, plus a little sound. Energy overall is still conserved; only mechanical energy is not. The general statement is:
Work done by non-conservative forces = ΔKE + ΔPE = change in mechanical energy
Here that work is −175 J, negative because friction opposes the motion. The same accounting explains a bouncing ball that returns to a lower height each time, a pendulum that eventually stops, and brakes that get hot.
The work-energy theorem
The net work done on a body equals the change in its kinetic energy: W_net = ΔKE = ½mv² − ½mu².
It follows in one line from Newton’s second law and v² = u² + 2as. Multiply that kinematic equation by m/2: ½mv² − ½mu² = mas = Fs = W. The theorem is more general than energy conservation, because it counts every force including friction, and it is often the fastest route to a speed when you know the forces and the distance.
A quick use: a 1000 kg car at 20 m/s brakes to rest in 40 m. ΔKE = 0 − 200,000 = −200,000 J, so the braking force is F = 200,000/40 = 5000 N, acting against the motion.
How mechanical energy differs from other forms of energy
| Form of energy | What it is stored in | How it relates to mechanical energy |
|---|---|---|
| Mechanical | Organised motion and position of a whole body | The reference case: KE + PE, directly visible and measurable |
| Thermal (internal) | Random motion of molecules inside the body | Microscopic and disordered, so it is not counted in ME; friction moves energy from ME into it and it cannot be fully moved back |
| Chemical | Bonds between atoms in fuel, food, a battery | Converts into mechanical energy in an engine or a muscle, with large losses as heat |
| Electrical | Charges moving under a potential difference | A motor turns it into mechanical energy; a generator does the reverse |
| Sound | Pressure waves in a medium | A small leak path for mechanical energy in impacts and vibration |
| Nuclear | Binding energy in the nucleus | Reaches mechanical form only after becoming heat, then steam, then turbine motion |
A hydroelectric plant is a clean chain to remember: gravitational PE of stored water becomes KE of falling water, which becomes rotational KE of the turbine, which the generator turns into electrical energy.
Power: how fast the energy is transferred
Energy says how much; power says how quickly. Power = work done ÷ time taken, P = W/t, in watts, where 1 W = 1 J/s. For a force acting on a body moving at speed v, P = Fv.
A pump lifting 50 kg of water through 10 m in 20 s does W = 50 × 10 × 10 = 5000 J of work, so its useful output power is P = 5000/20 = 250 W. The same job done in 10 s needs 500 W. The energy is identical; only the rate changes. One horsepower is 746 W, and one unit on a domestic electricity bill is 1 kWh = 3.6 × 10⁶ J.
References
FAQs
What is mechanical energy in simple words?
Mechanical energy is the total of an object’s kinetic energy and potential energy, ME = KE + PE. It is the energy the object has because it is moving and because of where it is placed or how it is stretched or compressed. It is measured in joules.
What is the formula for mechanical energy?
ME = KE + PE. Kinetic energy is ½mv², gravitational potential energy is mgh and elastic potential energy is ½kx². For a body moving at height h, ME = ½mv² + mgh. A 2 kg body at 5 m/s and 10 m high has 25 + 200 = 225 J, taking g = 10 m/s².
When is mechanical energy conserved?
Only when the forces doing work are conservative, which in practice means gravity and ideal spring forces. There must be no friction, no air drag, no sound losses and no external agent adding energy. Under those conditions KE + PE stays the same at every point of the motion.
What are some examples of mechanical energy?
A moving car, water stored behind a dam, a drawn bow, a swinging pendulum, a spinning flywheel, a hammer about to strike a nail and wind turning a turbine blade. Each has kinetic energy, potential energy or both.
Where does mechanical energy go when friction acts?
It becomes thermal energy in the rubbing surfaces, with a small part as sound. A 2 kg stone dropping 20 m should arrive with 400 J, and if friction leaves it with only 225 J, the missing 175 J has heated the surfaces. Total energy is still conserved; mechanical energy alone is not.