11.1 Motion of Objects

Calculus was invented by Isaac Newton who was trying to understand the motion of planets and their orbits around the sun, as well as the motion of anything else. Kepler had put together the astronomical data of that time, and had deduced his three laws from it.

Planets had elliptical orbits, Kepler had noted. He also found that the area swept out by the motion of planet (in the sort of triangle whose corners are the location of the sun and the position of the planet at two times separated by time dt) is a constant independent of where the planet is in its orbit. Finally, he found a relation between a measure of the size of the orbit, and the period of the motion around the sun.

Newton's aim was to construct a model for the motion of the planets that would lead to the results found from the data by Kepler.

He got the neat idea that a body, and that here means a planet, would keep on going straight ahead at constant speed, if left strictly alone. So that the derivative of the planet's position with respect to time, which is its velocity, would be constant, if the sun did not act on it.

Another way to say the same thing is that in the absence of the sun, the derivative of the velocity of the planet, which derivative is called the acceleration of the planet, would be zero.

He developed the idea that the sun exerts a "force" on the planet, which force changes the acceleration of the planet in proportional to the force.

He was able to show that if the force on the planet was directed toward the sun, and was proportional to the reciprocal of the square of the distance of the planet to the sun, the resulting expression for the second derivative of the position of the planet could be solved, and the equations have solutions in which the orbit of the planet is elliptical and equal areas are swept out by the planets motion in any time interval of given size.

He also showed that if the force on the planet was proportional to the "mass" of the planet, (which is the factor relating the force to the acceleration: as in F = ma, with F the force, m the mass of the object and a its acceleration, the second derivative of its position) then the third of Kepler's laws also holds. This led Newton to his law of gravity: that bodies attract one another with a force proportional to the masses of each and inverse to the square of their distance.

We will not discuss the details of Newton's argument here, because the motion is in space, and position and velocity and acceleration require more than one number to describe each of them, so that we would first have to discuss how to define how to represent these entities.

He also showed that the motion of objects thrown or dropped could be deduced, in this model, from the almost constant gravitational attraction of the earth on the objects.
The observation of Galileo that objects of different weights fall, to a first approximation (ignoring the effects of air resistance) at the same speeds, also follows from Newton's law of gravity, among many other things.

If we represent the height of an object from some given base line as h(t) with the argument t given by the time, starting at some given time, the effects of the earth's gravitation on it will obey the equation

where g is a constant, called the gravitational constant, and m is the mass of the object.

This is the equation for the behavior of a falling object according to Newton's model of gravitation.

In this particular case, we can deduce the consequences of this model quite easily.

This tells us that the acceleration, which is the derivative of the speed of the object, is constant. What functions have a constant -g as their derivative?

You will recall that taking the derivative of a polynomial lowers the power of each of its terms by 1, and multiplies it by the power. The answer we seek is: any function of the form -gt + c will have -g as its derivative with respect to variable t.

The next question is, what function, h(t) has -gt + c as its derivative?

The answer to this is , where d, like c is a constant that you must determine from the initial conditions on your specific problem.

In general, the plan for describing a phenomenon is to emulate Newton:
First find some equations involving derivatives with which to describe the behavior of the system.
Then work out the consequences of the model, by working backwards from the derivatives to the function itself.

And why bother with derivatives? Because the behavior of systems in their linear approximation is much much simpler than their ordinary behavior after a reasonable length of time. Creating models for the behavior of derivatives, here of second derivatives, is incredibly easier than creating direct models for general behavior.

Even in this simple case, the model for the second derivative involves only a universal constant, g, while to model height versus time directly requires two additional parameters and a quadratic function.