Our author sees conservation of energy as the single most important principle in all of physics (or mechanics at least).
In ordinary interactions an accounting can be made of all the energy that exists in the system prior to the interaction and all the energy in the system after the interaction. If in each instance all energies are accounted for the two totals will be the same.
In real interactions it's very difficult to retain all energy within a defined system (heat is very hard to keep track of). It may also be hard to quantify energy changes that do stay within a system (the potential energy change that results from deformed sheet metal in an auto collision for instance). We'll keep to ideal systems that permit us, with care, to keep track of all the energy of a system.
Gravitational Potential Energy:
The energy that an object has as a result of its position (height) in space. When considered with respect to the usual definition in terms of the capacity to do work, what we mean is how much work an object can do while it falls to earth (or to some other reference height). This energy is given by PE = mgh where PE is potential energy (often U in advanced courses) m is mass, g is the local acceleration of gravity and h is the height. In the SI system the energy will be given in Joules when the mass is in kilograms, the height in meters and the value of g taken as 9.8 m/s/s (read "meters per second squared").Kinetic Energy:
The energy posessed by a mass as a result of its motion. The amount of energy is directly proportional to the object's mass and to the square of its velocity, thus for a moving object, doubling its mass will only double its energy where doubling its velocity actually quadruples its energy. KE = 1/2 mv^2.Elastic potential energy. . .
takes various forms. We'll keep it simple and consider only "perfect" springs. Real-life elastics such as rubber bands and the coils from demolished ball-point pens are examples of imperfect approximations of these rules. In a perfect spring if we pull or push twice as hard the spring will compress or stretch twice as much -- exactly. The quotient of the force used to stretch the spring and the amount of stretch that results is thus a constant -- called the spring constant or the Hooke's Law constant.The energy in a stretched or compressed spring is given by PE = 1/2 kx^2 where k is the spring constant in newtons per meter and x is the amount of stretch (or compression) in meters.
Work
If energy is the capacity to do work then work is the result of energy consumed. When a stretched rubber band returns to its unstretched length it does work on the stone placed in the sling shot. The same energy is now the kinetic energy of the stone. If the stone is fired straight up the velocity and thus the kinetic energy is decreased, while the height and thus the gravitational energy is increased. At the highest point the kinetic energy is 0 and the potential energy is maximum -- and, if we had perfect conversions, equal to the energy stored in the band before release.
Conservation of Mechanical Energy
We will here consider only conservation of mechanical energy. For a system not affected by friction or air resistance we can sum all the mechanical energy before an interaction and expect it to be the same after the interaction. However, the value of each term can change. The only terms that we'll consider are kinetic, gravitational potential and elastic.
For one single object in a system we have (where the accent mark is read "prime" and signals the "after" version of a quantity so marked):
For two objects interacting (exchanging) energy we could have:
It's easy to see how complicated things get when several or several hundred objects interact.
Often one or several of these terms is zero. For instance the kinetic energy (Ek) is zero for any object at rest. If there is no spring in the problem there is no elastic energy term on either side. Most problems become much simpler once the zero terms are identified.