Examples in classical mechanics often involve things like hockey pucks on ice and balls being thrown through the air, because these things are relatively easy to make very accurate predictions about.
The majority of classical mechanics can be derived from Newton's Laws of Motion, and since they (particularly the second) concern rates of change of properties, classical mechanics was one of first branches of applied mathematics to make use of Newton's own concepts from calculus, in particular casting physical problems in terms of differential equations.
For example, one of the fundamental concepts of classical mechanics is velocity, or the rate of change of position with time i.e.
The strength of a gravitational field is given in terms of acceleration. Since mass is not involved in this equation, that means that two objects dropped from the same height will land at the same time, assuming that there there is no friction (during one of the moon missions this was dramatically shown by an astronaut dropping a hammer and a feather. Both hit the lunar surface at the same time).
Acceleration can be achieved in two ways: changing speed, and changing the direction of motion. A good example of changing speed is to simply drop a pencil. The speed of the pencil will increase by about 10 m/s each second. Slowing down (often called decceleration) is, in terms of physics, still called acceleration because you are still accelerating, just in the direction opposite of your motion. Changing the direction of motion is a less intuitive example of acceleration, but just as valid. A good example of this is a circular freeway onramp: the car is moving at constant speed, but the passengers are squished to one side. The car is accelerating them, or changing their direction, to keep them from moving in a straight line.
Classical mechanics then, deals with the relationship between position, velocity, acceleration, mass, and force, and these are related by Newton's second law:
However, a more useful representation is F/m = a, or an acceleration results when a force is applied to a mass.
Rockets in space are usually used as examples for this equation, because it isn't particularly intuitive here on earth with friction. Some specific rocket booster will exert a certain amount of force on the rest of the rocket. If that booster is pushing a rocket with a small mass the rocket will have a certain acceleration. If it is pushing ten times that small mass, it will have one tenth that acceleration.
Another important concept in classical mechanics is work.
When a force is applied to a body over a given distance, the result is work. For example, for the case of the rocket mentioned above, if a force of 10 N is applied over a distance of 200 m, the resulting work is, W= 10N*200m = 2000 J. In this case the work will result in a change in the velocity of the rocket.
Energy is defined as the ability to do work.
Energy can exist in a number of forms. Of particular interest for the field of Mechanics is Kinetic energy. Kinetic energy is that portion of energy associated with the motion of a body.
For the case of the rocket mentioned above, the 2000 J of work on the rocket would result in an increase of the kinetic energy of the rocket by 2000 J.
Classical mechanics well describes the behaviour of most familiar objects. Only when looking at a very small scale, or at objects moving very fast, are different theories necessary - respectively Quantum Mechanics and Relativistic Mechanics (see Theory of Relativity).
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