Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

The material of Part A has been gathered together to provide an overview for the budding aeronautical engineer and rocket scientist. Part B will focus on the numerical solution of related differential equations.

Known early attempts to gain an understanding of gravity began with Aristotle's belief that for every effect there is a cause. Aristotle believed that heavier objects accelerate faster in falling than do lighter objects.

About 1000 years later Galileo held that

Galileo's work was quickly followed by Newton's theory of gravitation.

Kepler, a mathematician hired to calculate for an astronomer Tycho, is thought to have inspired Newton's work. An excerpt from here follows:

1.

2. As the planet moves along its path, a line joining the planet to the Sun sweeps out equal areas in equal times.

3. For any two planets, the ratio of the squares of their periods of revolution about the Sun is the same as the ratio of the cubes of their mean distances from the Sun.

"

Although Newton's hypothesis stimulated many astronomical advances including the discovery of the planet Neptune, it failed to satisfactorily account for the orbit of Mercury. The failure was first explained by Einstein's general relativity theory and later by a development founded in classical physics that avoided the need to employ relativity theory. See:

Today, most earthly and astronomical gravitational calculations are based on Newton's work because it is an easier theory to work with and is sufficient for most applications.

Newton's theory of gravitation and his three laws of motion have led to evaluation of the mass of our sun, of its planets, and of the distances to stars.

Projectiles, as a topic in ballistics, provides an interesting discussion on drag and on experimental methods.

Fluid Mechanics provides the conclusion that the power required to overcome drag is proportional to the cube of velocity. See Drag Power.

For higher velocities there is an equation attributed to Lord Rayleigh that proposes the drag force as proportional to the square of an object's velocity. His approximation states:

The drag coefficient,

The density of our air varies in accord with height above sea level. For equations and values see Barometric formula. Another interesting presentation can be found here.

It gets quite complicated. Is the fluid compressible? What is the effect of buoyancy? Or of lift? Much care is advised for the use of any of the many approximations proposed for characterizing the motion of a body in a fluid.

Lord Rayleigh's approximation to the drag force,

vt = ((2 * m * g)/(

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