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Up to this point, physics has been all fun and games, right? Well, time to roll up your sleeves and get down to business. You see, the energy of an object is its ability to do mechanical work. When an object does work (W), we mean it is exerting a force (F) over a distance (d):
W = Fd
One caveat: the direction the force is applied and the direction of motion must be the same in order to use this formula. If they're not—such as a car driving up a hill—you have to look at the component of the force that's in the direction of motion (and here you probably thought you were done with all that component stuff after the last chapter, huh?).
The unit of work is a newton-meter, or a kg⋅m2/s2—exactly the same as a joule. When a force acts over a distance, it changes the kinetic energy of the object it acts on. This is known, creatively, as the work-energy theorem and can be expressed by an equation.
Here vf is the object's final velocity and vi its initial velocity.
Lots of objects we see every day are doing work all the time—think of, for example, a crane lifting a beam from the ground to the top of a building under construction. The crane must move a force (the weight of the beam) some distance (the height of the building), and in doing so changes the beam's kinetic energy (from zero when the beam's at rest on the ground to nonzero as it's raised upwards).
But what about conservation of energy? We can't create energy, after all, but if we're lifting a beam, it's getting energy from somewhere. The answer is the same as in our discussion of momentum—in the case of the beam, the crane's motor is outside our system, and so the force doing the work to lift the beam is an outside force.
What's important to note here, though, is a distinction between kinds of outside forces. Forces like the one coming from the crane are what we call non-conservative forces; they don't conserve the mechanical energy of the system. They can either add to the system's energy—moving energy from the crane's gasoline to the beam—or take energy away from the system, like friction would do to a rolling bicycle.
Some forces, however, are conservative. They conserve the mechanical energy of our system. Conservative forces usually act based on an object's position. Gravity, for example, can convert an object's potential energy to kinetic, but must keep the total amount of energy the same.
When in doubt, remember the story of Icarus. Icarus straps on some sweet wax wings courtesy of Papa Daedalus. He gets all sorts of lift, soaring up in the air and giving him a ton of kinetic and potential energy (the aerodynamic force from the air is definitely non-conservative). Once he gets too high, though, the sun melts his wings. Suddenly the only force acting on poor Icarus is cold, conservative gravity. That means all the potential energy—and there's a lot of it—must be converted to kinetic energy. All that kinetic energy makes for a pretty bad ker-splat.
Morals of the Icarus story: 1) you gotta make your wings out of carbon fiber, not wax, bro, and 2) pay attention to how outside forces can change the form of energy of an object. Don't forget that forces like friction are going to remove mechanical energy from an object, while other outside forces can add mechanical energy in the form of either kinetic or potential energy.
We've been in suspense since the introduction over seatbelts. Now we know enough physics to figure out why they're so effective at preventing serious injury. In a car collision, one of the worst things that can happen is smacking your head against the car's dash. This brings you to a stop almost instantly, changing your kinetic energy from to zero in a very, very small distance, d.
From the work-energy theorem, we know the force you'd feel in such a collision is , and when the dash is stopping you instantly, that means the force is huge. Generally, big forces and squishy human bodies don't play particularly well together.
The seatbelt alleviates a lot of this problem by stopping you more gradually—the seatbelt stretches some nonzero distance, d, and suddenly the force you feel for the same loss of kinetic energy is much, much smaller.