Work Equals the Change in Kinetic Energy
Push a box with a constant force and move it some distance. Force times distance is the work. For the same force, pushing farther is more work; for the same distance, pushing harder is more work. But where does this work go? With no friction, the work you do turns entirely into the box's kinetic energy: the more work you do, the faster the box goes. This is the work-energy theorem. The work done by the net force equals exactly the change in kinetic energy, W = ΔKE. Pair force with time and you get momentum; pair force with distance and you get energy. It is another way of seeing the same Newton's law, accumulated over space instead of time.
Push a box a distance d with a constant force F. The work done is W = F·d, force times distance. In the graph below, the horizontal axis is the distance traveled and the vertical is the force, and the work is the area of the shaded rectangle beneath. Drag d and the area grows in proportion to the distance. The unit is N·m, which we call the joule (J). One caution: work counts only the component of force along the direction of motion. If the force is perpendicular to the motion (such as the centripetal force in circular motion), the work is zero. Even when you hold something and stand still, the upward force does no work, because there is no distance moved. Work is how much the force has actually pushed along the distance.
Where does the work go? On a frictionless floor, the work done by the net force goes entirely into kinetic energy. Do an extra amount of work W on a box that already has some kinetic energy, and the kinetic energy grows by exactly that much. Raise the work W with the slider. The kinetic-energy bar grows by W, and the speed rises accordingly. As a formula, W = ΔKE = KEfinal − KEinitial. This is the work-energy theorem. The nice part is that you do not need to know how the force changed along the way or how much time it took: just look at the kinetic energy at the start and the end and you immediately know the work the net force did. Doing negative work (opposite to the motion, like braking) shrinks the kinetic energy and slows it down.
Let us see exactly what kinetic energy is. The kinetic energy an object of mass m has while moving at speed v is KE = ½mv². Drag the v slider and the kinetic energy grows as the square of the speed. Double the speed and the kinetic energy is four times as much. This pairs with the v² = v₀² + 2a·Δx we saw earlier: multiply both sides by ½m and out comes the work-energy theorem itself. The fact that speed enters as a square bites hard in real life. Double a car's speed and its kinetic energy is four times as large, so the work needed to stop it is four times, and the braking distance is four times. That is why collision damage is so sensitive to speed.
Force is not always constant. Even so, work is always found the same way: 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 stretching a spring (F = kx), the area is a triangle, so W = ½kx². For a force that fades away it is yet another shape. Whatever the shape, work is the gathered area under the curve, that is, the force integrated over distance. This is the very same integration as in the earlier lesson, where we integrated velocity over time to get distance; we just accumulate over distance instead of time. Even a varying force, where simple multiplication fails, is always solved by reading the area.
Finally, power. Doing the same work but finishing faster is more powerful. Power is the work done per unit time, P = Wt. And if you are moving at speed v under a constant force, power can be written cleanly as P = F·v. Drag the v slider and, for the same force, the faster you go the greater the power. The unit is the watt (W); one watt is doing one joule per second. An elevator lifting the same weight to the same height needs a more powerful motor to do it quickly, and a car needs a stronger engine to deliver the same force at higher speed. If work is the total amount of energy, power is the rate at which it is poured out. In the next lesson we widen this energy to include potential energy and move to the conservation of energy, where, as long as only conservative forces act, the sum of kinetic and potential energy stays constant.