work-energy theorem explained
Push a box with a steady force and move it some distance. Force times distance is work, so the same force pushed farther does more work, and the same distance pushed harder does more work too. Where does that work go? Without friction, all the work we do becomes the box's kinetic energy, and the box speeds up by exactly as much work as it received. This is the work-energy theorem: the work done by the net force equals the change in kinetic energy (W = ΔKE). Accumulating force over time gives momentum, but accumulating it over distance gives energy; it is the same Newtonian law seen across space rather than across time.
Push a box with a constant force F (newtons) through a distance d (meters) and the work done is W = F·d, force times distance. In the graph, distance traveled runs along the horizontal axis and force up the vertical, and the work is the area of the gold rectangle under the orange force line. Increase d with the slider and that area widens in proportion to the distance, while the work in the readout grows with it. The unit is the N·m, called the joule (J). Work counts only the component of force along the direction of motion, so a force at right angles to the motion (like the centripetal force in circular motion) does 0 work. Standing still while holding something heavy, there is no displacement, so the upward supporting force does no work. Work is how much a force actually pushes along a distance.
Where does the work go? On a frictionless floor, the work W (joules) done by the net force goes entirely into kinetic energy. The box in the figure starts with some kinetic energy already; increase the work done with the slider and the kinetic-energy bar grows by the gold W segment attached to the blue initial part, while the blue velocity arrow lengthens. As W rises, the kinetic energy rises by exactly that amount; in symbols, W = ΔKE = KEfinal − KEinitial, the work-energy theorem. Its usefulness lies in not needing to know how the force varied along the way or how long it took: the kinetic energy at the start and end alone tells you the work done by the net force in between. Negative work, against the motion as a brake does, lowers the kinetic energy and slows the box.
Now look at what kinetic energy is. It depends on two quantities, the mass m and the speed v, and is written KE = ½mv². Drag the velocity slider v and the gold bar grows with the square of the speed, so doubling the speed makes the kinetic energy four times larger. This formula pairs with v² = v₀² + 2a·Δx from earlier: multiply both sides of that by ½m and out comes the work-energy theorem. The fact that speed enters squared has frightening consequences in everyday life. A car going twice as fast has four times the kinetic energy, so the work needed to stop it and its braking distance are both four times larger; that is why collision damage is so sensitive to speed.
Even when the force is not constant, work is found the same way: it is the area under the F-x graph. Switch the shape of the force with the buttons. For a constant force the area is a rectangle, so W = F·d. For a force that grows in proportion to distance, like the pull of a spring (F = kx), the area is a triangle and W = ½kx²; a force that dies away gives yet another shape. Whatever the shape, work gathers up the gold area under the curve; it is the integral of force with respect to distance. It is the same integral that turned velocity into distance in an earlier lesson, taken over distance instead of time, so even a varying force that a single multiplication cannot handle always yields to the area.
Last comes power. Doing the same work faster is more powerful, and power is work per unit time, P = Wt. If a constant force keeps an object moving at speed v, the power is neatly P = F·v. Drag the velocity slider v and, with the force unchanged, the gold power bar lengthens the faster the object goes. The unit is the watt (W), and 1 watt means doing 1 joule of work in 1 second. An elevator lifting the same weight to the same height needs more power if its motor lifts faster, and a car that must deliver the same force at higher speed needs a stronger engine. If work is the total amount of energy, power is the rate at which it is poured out. The next lesson extends energy to potential energy and arrives at energy conservation: with only conservative forces, kinetic plus potential energy stays constant.