Chapter 4

Falling Bodies In a Fluid or Gas Part A - Gravity - Fluid Mechanics - Aerodynamics - Differential Equations

The topic of falling bodies may sound simple; we see things falling all the time. It is not simple. It is very complex. The culprit is resistance to gravity.

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.

Gravity

See Gravity for an article.

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 air resistance was the reason for Aristotle's belief that heavier objects accelerate faster in falling and that without this resistance all falling bodies would accelerate at the same rate.

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:

Johannes Kepler (1571 - 1630) is now chiefly remembered as a mathematical astronomer, in particular for discovering three laws that describe the motion of the planets. In their modern forms, these are

 1.  The path of each planet is an ellipse with the Sun at one of its foci.

 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.

These are usually known as "Kepler's three laws of planetary motion.

"It was from the third law that Isaac Newton (1642 - 1727) was to deduce the existence of an inverse square law of gravitation."

(From Mathematical Principles of Natural Philosophy or, to give it its Latin title, Philosophiae Naturalis Principia Mathematica, London, 1687.)

Newton's Hypothesis

The cornerstone of that theory is Newton's hypothesis of an inverse square relationship, in Newton's words:

  “I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly.”

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: Kidman, J. N. (1977). Quantum gravitation and the perihelion anomaly. Lettere al Nuovo Cimento, 18, pp. 181-182.  See here for a PDF file of the article.

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.

Drag

When a body moves through a fluid, gas or liquid, there are forces, that resist such motion. These forces together form Drag.  Drag Force increases with the density of the fluid, ρ, a value A closely related to the area of the body, and a shape related drag coefficient, Cd .

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.

Unsolved

The theoretical determination of drag forces is an unsolved problem of hydrodynamics.  Most available data has been arrived at by experiment.  Dimensional analysis (See Classical Mechanics, R. Douglas Gregory, Cambridge University Press, 2006, pp 82-83.) indicates that the drag force in many practical instances must be proportional to the square of velocity.

Practical Approximations

Stokes's Drag is an equation that may be appropriate for particles that are moving slowly, creeping, through a viscous fluid. It states that the resistance is approximately proportional to the velocity of the particle.

(An introduction to Absolute Viscosity and Kinematic Viscosity can be found here.)

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:

                 Fd = 1/2 * ρ * A * Cd* v2

An explanation for these two diverse views is believed to be that turbulence occurs when the inertial properties of the medium are more dominant in the motion than the viscous properties. In about 1893 Osbourne Reynolds popularized a measure of the ratio of inertial forces to viscous forces that is called the Reynolds number. See Reynolds number.

The drag coefficient, Cd, is body shape related.  Wind tunnels have been used to obtain much of the information on drag. Aerodynamic drag coefficients for a range of body shapes can be found in " Drag of Cylinders & Cones".  For some shapes the data is treated as proprietary.

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. 

Potential Energy

There is a change in potential energy when a change of height occurs. For a small range of height over which g is taken as constant, this is calculated as m * g * h or w * h where the weight w, a force, is the product of the mass of a body and the local gravitational acceleration value, and h is the height.

Terminal Velocity

For a falling body, drag opposes the acceleration of its mass and a point of equilibrium is reached when there is no further increase in velocity, i.e., terminal velocity.

Lord Rayleigh's approximation to the drag force, Fd = 1/2 * ρ * A * Cd* v2, and Newton's second law, F = m * a, can be employed, (by setting drag equal to the gravitational force mg), to find an expression for terminal velocity:

                               vt =  ((2 * m * g)/(ρ * A * Cd))1/2 

Differential Equations for Fall

When the drag is proportional to the square of velocity, the differential equation is not linear. For the case when the multiplier of v2 is small, an approximate solution in PDF form can be found here.

Next

Numerical solution of differential equations for falling bodies will be treated in Part B of this topic.

(For numerical methods for solving differential equations see here.)

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