Energy, work, and power get mixed up constantly, mostly because everyday English uses them almost interchangeably. Each one has a distinct, precise definition in physics, and together they unlock a huge range of practical calculations: crash safety, engine ratings, why a longer wrench makes a bolt easier to loosen. This guide walks through six related concepts in sequence, building from the two core forms of mechanical energy up through work, the rate at which work is done, and two rotational and circular-motion concepts that extend the same ideas to spinning and curving objects.
Each step includes a worked numeric example and a link to the calculator that automates the underlying formula, so you can check a hand calculation or see how changing one variable, mass, height, speed, radius, shifts the result.
Step 1: Kinetic Energy (Energy of Motion)
Kinetic energy is the energy an object has because it's moving, calculated as KE = ½mv², where m is mass in kilograms and v is velocity in meters per second. Because velocity is squared in this formula, kinetic energy grows much faster than speed itself. Doubling speed quadruples kinetic energy, and tripling speed multiplies it by nine.
Worked example: A 1,500 kg car traveling at 25 m/s (about 56 mph) has a kinetic energy of KE = 0.5 × 1,500 × 25² = 0.5 × 1,500 × 625 = 468,750 joules. If the same car speeds up to 35 m/s, its kinetic energy becomes 0.5 × 1,500 × 1,225 = 918,750 joules, nearly double, even though speed only increased by 40%. The Kinetic Energy Calculator computes this instantly for any mass and velocity, and it's especially useful for showing why small increases in speed disproportionately raise crash energy.
Step 2: Potential Energy (Stored Energy)
Potential energy is stored energy based on an object's position rather than its motion, most commonly gravitational potential energy, calculated as PE = mgh, where m is mass, g is gravitational acceleration (9.8 m/s² on Earth), and h is height above a reference point. This energy is called "potential" because it has the capacity to convert into kinetic energy, for instance as an object falls.
Worked example: A 5 kg object sits on a shelf 2 meters above the floor. Its potential energy relative to the floor is PE = 5 × 9.8 × 2 = 98 joules. If it falls, all 98 joules converts into kinetic energy just before impact (ignoring air resistance). You can verify this by using the Potential Energy Calculator to find the 98 joule figure, then checking that the corresponding fall velocity from the Free Fall Calculator produces the same kinetic energy value when plugged into the Kinetic Energy Calculator.
Step 3: Work, Force Acting Over Distance
In physics, work has a strict definition that differs from its everyday meaning. Work is done only when a force causes displacement, calculated as W = Fd, force multiplied by distance, when force and displacement point the same direction. Pushing against an immovable wall, no matter how hard, does zero physics-defined work, because nothing displaces.
Worked example: Lifting a 20 kg box a vertical distance of 1.5 meters requires a force equal to its weight (20 × 9.8 = 196 N) applied over that 1.5 meter distance, so the work done is W = 196 × 1.5 = 294 joules. This matches the potential energy gained by the box (PE = mgh = 20 × 9.8 × 1.5 = 294 joules): work done against gravity converts directly into gravitational potential energy. The Work Calculator computes this from any force and distance combination.
Step 4: Power, The Rate of Doing Work
Power measures how quickly work gets done or energy transfers, calculated as P = W/t, work divided by time, with the standard unit being the watt, one joule per second. Two tasks can involve identical total work, but the one finished faster requires more power.
Worked example: Using the 294 joules of work from Step 3, if one person lifts the box in 2 seconds, their power output is P = 294/2 = 147 watts. If another person takes 6 seconds to lift the identical box the identical height, their power output is only 294/6 = 49 watts, a third as much, despite doing the exact same amount of work. The Mechanical Power Calculator computes power from any work-and-time or force-and-velocity combination, and it can convert results to horsepower for comparing against engine or motor ratings.
Step 5: Torque, Rotational Force
Torque is the rotational equivalent of force. It measures a force's tendency to cause rotation around a pivot point, calculated as τ = Fd, where F is the applied force and d is the length of the lever arm, the perpendicular distance from the pivot to where the force is applied. Longer lever arms produce more torque for the same applied force, which is why longer wrenches and longer door handles take less effort to use.
Worked example: Turning a bolt requires 80 N of force applied at the end of a 0.25-meter wrench, producing a torque of τ = 80 × 0.25 = 20 N·m. Switching to a 0.5-meter wrench, twice as long, would need only 40 N of force to produce the same 20 N·m of torque: half the effort for the same rotational result. The Torque Calculator solves for torque, force, or lever arm length given the other two values.
Step 6: Centripetal Force in Circular Motion
Centripetal force is the net force required to keep an object moving along a circular path, always directed toward the center of the circle, calculated as F = mv²/r, where m is mass, v is speed, and r is the radius of the circular path. Without this inward force, an object would fly off in a straight line, per Newton's first law, rather than curving.
Worked example: A 1,000 kg car takes a curve with a 50-meter radius at 15 m/s. The centripetal force required is F = (1,000 × 15²)/50 = (1,000 × 225)/50 = 4,500 N, supplied by friction between the tires and the road. If the car speeds up to 20 m/s on the same curve, the required force jumps to (1,000 × 400)/50 = 8,000 N, nearly double, which is why taking a curve too fast can exceed the available tire friction and cause a skid. The Centripetal Force Calculator computes the required force for any mass, speed, and radius combination.
Putting It Together: Conservation of Energy
A theme runs through all six of these concepts: the principle of conservation of energy. Energy is never created or destroyed, only transformed from one form to another, kinetic to potential, work into stored energy, and so on. A pendulum swinging back and forth continuously trades kinetic energy for potential energy and back. A car's engine converts chemical energy in fuel into work done against friction and air resistance, delivered at a certain rate, power. A spinning object's rotational motion is governed by torque in exactly the same way linear motion is governed by force.
Understanding where energy is stored, how quickly it transfers, and what forces act rotationally or circularly gives you the tools to reason about a huge range of everyday physical situations, from why a roller coaster doesn't need an engine after the first hill, to why a highway curve has a speed limit, to why a socket wrench with a longer handle feels easier to use.
Common Mistakes to Avoid
The most frequent error is treating "work" as synonymous with effort. Physics only counts work when there is actual displacement in the direction of the applied force, so holding a heavy box stationary, no matter how tiring, involves zero work by the strict definition. A second common mistake is confusing energy with power: a small motor and a large engine can deliver the exact same total energy over time, but the large engine does it faster, meaning higher power, not necessarily more total work. A third mistake is forgetting that torque depends on lever arm length as well as applied force. Pushing on a door right next to its hinge takes far more force to open than pushing at the handle, even though the force itself hasn't changed, because the lever arm has shrunk. Students also often assume centripetal force is a separate force pushing outward. It isn't. It's simply the name given to whatever net inward force, gravity, tension, friction, or some combination, is actually keeping an object on its circular path. There is no separate "centrifugal force" acting on the object itself, just the sensation of being pushed outward due to inertia.
How These Concepts Connect in Practice
A roller coaster illustrates nearly every concept in this guide at once. At the top of the first hill, the cars have maximum potential energy and minimal kinetic energy. As they descend, that potential energy converts into kinetic energy, following the conservation principle from the section above. The motor that pulled the cars up the hill did work against gravity, and the rate at which it did so was its power output. On any curved section of track, centripetal force, supplied by the track pushing on the cars, keeps them following the curve instead of flying off tangentially. Any rotating component in the lift mechanism, like a gear or pulley, is governed by torque rather than force alone, since it's turning rather than moving in a straight line.
This same pattern, potential and kinetic energy trading back and forth, work being done at some rate, torque or centripetal force governing anything that rotates or curves, shows up across engineering and everyday physics alike, from car engines and wind turbines to something as simple as a child on a swing.
Key Terms
- Kinetic Energy: the energy an object possesses due to its motion, equal to ½mv²
- Potential Energy: stored energy based on an object's position, most commonly gravitational potential energy (mgh)
- Torque: the rotational equivalent of force, equal to force multiplied by lever arm length
- Work: energy transferred by a force acting over a distance, measured in joules
- Power: the rate at which work is done or energy is transferred, measured in watts
- Centripetal Force: the net inward force required to keep an object moving along a circular path
- Conservation of Energy: the principle that total energy in an isolated system remains constant, only changing form