Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Kinetic EnergyThe technical term for the energy associated with motion is kinetic energy, from the Greek word for motion. (The root is the same as the root of the word cinema for a motion picture, and in French the term for kinetic energy is énergie cinétique.) To find how much kinetic energy is possessed by a given moving object, we must convert all its kinetic energy into heat energy, which we have chosen as the standard reference type of energy. We could do this, for example, by firing projectiles into a tank of water and measuring the increase in temperature of the water as a function of the projectile's mass and velocity. Consider the following data from a series of three such experiments:
Comparing the first experiment with the second, we see that doubling the object's velocity doesn't just double its energy, it quadruples it. If we compare the first and third lines, however, we find that doubling the mass only doubles the energy. This suggests that kinetic energy is proportional to mass and to the square of velocity, KE mv^{2}, and further experiments of this type would indeed establish such a general rule. The proportionality factor equals 0.5 because of the design of the metric system, so the kinetic energy of a moving object is given by
The metric system is based on the meter, kilogram, and second, with other units being derived from those. Comparing the units on the left and right sides of the equation shows that the joule can be reexpressed in terms of the basic units as kg·m^{2}/s^{2}. Students are often mystified by the occurrence of the factor of 1/2, but it is less obscure than it looks. The metric system was designed so that some of the equations relating to energy would come out looking simple, at the expense of some others, which had to have inconvenient conversion factors in front. If we were using the old British Engineering System of units in this course, then we'd have the British Thermal Unit (BTU) as our unit of energy. In that system, the equation you'd learn for kinetic energy would have an inconvenient proportionality constant, KE = (1.29 × 10^{3}) mv^{2}, with KE measured in units of BTUs, v measured in feet per second, and so on. At the expense of this inconvenient equation for kinetic energy, the designers of the British Engineering System got a simple rule for calculating the energy required to heat water: one BTU per degree Fahrenheit per gallon. The inventor of kinetic energy, Thomas Young, actually defined it as KE = mv^{2}, which meant that all his other equations had to be different from ours by a factor of two. All these systems of units work just fine as long as they are not combined with one another in an inconsistent way.
Is there any way to derive the equation KE = (1/2)mv^{2} mathematically from first principles? No, it is purely empirical. The factor of 1/2 in front is definitely not derivable, since it is different in different systems of units. The proportionality to v^{2} is not even quite correct; experiments have shown deviations from the v^{2} rule at high speeds, an effect that is related to Einstein's theory of relativity. Only the proportionality to m is inevitable. The whole energy concept is based on the idea that we add up energy contributions from all the objects within a system. Based on this philosophy, it is logically necessary that a 2kg object moving at 1 m/s have the same kinetic energy as two 1kg objects moving sidebyside at the same speed.


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