Michael Lee's Mathematics and Physics
Derivation of Newton's Kinetic Energy
Newton defined energy as that which is capable of doing work. Furthermore, he defined work as the force applied to a body multiplied by the distance travelled (displacement) of the body while the force is applied. Work done is actually its change in kinetic energy, but since we are going to assume the body's initial velocity is zero, work done is equal to the body's kinetic energy; it's energy of motion as opposed to other kinds of energy like potential energy for example.
According to Newton's Second Law of Motion, the acceleration of a body, as produced by a net force, is directly proportional to the magnitude of the force applied to it and inversely proportional to the body's mass.
Substituting Newton's Second Law into his equation for work results in the following:
To remember, just think, "doing work is madness."
I may as well mention it now. Newton's first law (the law of inertia) says a body at rest stays at rest and a body in motion stays in motion with uniform velocity (magnitude and direction) unless acted upon by a net force. Newton's third law says a force applied to a body, in turn, induces a force of equal magnitude and opposite direction as the original force. For example, a guitarist applies pressure to the strings to adjust its pitch; the strings, in turn, apply forces back causing his or her finger tips to become tender and swollen after playing too long. The mass and acceleration of a body are usually given so all that remains is determining what the distance travelled by the body is; in other words, its displacement. For uniform velocity (no net force), a graph of velocity vs time looks like this:
For the above graph, we have uniform velocity and so the acceleration is equal to zero, thus there is no net force being applied to the body. Notice how the slope of the line Velocity = V is zero (in maths it is the y axis). Time (T) is on the x axis in maths. Thus, displacement d is simply the area under the line and notice it's in the shape of a simple rectangle. For example, if a car is travelling at a constant velocity of 100 Km/hour and it travels for four hours, the distance travelled or displacement of the body is simply:
In the graph above, notice how the displacement of a body is equal to the area under the line of Velocity = V, but what if the body is accelerating as a result of a net force being applied to it? Well, acceleration is simply the rate of change of velocity over time. In other words, the line would still be linear but with a slope equal to the acceleration of a body under the application of a uniform force. Under uniform force the graph would look like this:
Just as with uniform velocity, the displacement of the body is equal to the area under the line. However, notice how it is now in the shape of a right triangle. Therefore, the body's displacement under uniform force and hence uniform acceleration, is equal to:
Newton's Calculations can be taken one step further to obtain the following...
