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This paper provides an enjoyable example through which several concepts of classical mechanics can be understood. We introduce an equation that models the motion of a cat in the presence of a person. The cat is considered as a point particle moving in a potential induced by the person. We demonstrate that this approach to the problem reproduces characteristic behaviors of these curious animals. For instance, the fact that cats do not typically come when they are called, or that they remain longer on the lap of their favorite person; even "zoomies" are reproduced (cats randomly run back and forth across the house). We use this model problem to explore topics of current research such as stochastic equations and periodically driven systems. The pedagogical value of this work and its potential use in teaching are discussed. Editor's note: Since Erwin Schrödinger, cats have become the unofficial mascots of quantum mechanics. But cats can be useful in classical mechanics too. This paper shows that typical feline behaviors, such as their preferred resting position (on your lap or, more likely, far away), ignoring calls, and purring, can actually be explained by classical mechanics. Even zoomies, those periods of time when cats are frantic, can also be modeled as a stochastic process. So, it turns out your cat might just be the perfect example for most of an undergraduate mechanics course! [ABSTRACT FROM AUTHOR]

Among the several methods to compute the perihelion precession for bounded orbits in Schwarzschild spacetime, the simplest is to ignore a term in the equations of motion. This is currently justified under the assumption that the eccentricity of the orbit is small. For cases such as Mercury in our solar system, whose eccentricity is not small, this method seems not to be applicable. Yet it gives the right result, the reason being that the term that has been excluded, although responsible for first order—in the ratio of the Schwarzschild radius over the radial coordinate—corrections of the orbit, only produces completely negligible higher order corrections for the perihelion precession. We show this result by two different procedures. We claim, therefore, that as long as the aim of the computation is the perihelion precession, one can safely drop that term regardless of the magnitude of the eccentricity. [ABSTRACT FROM AUTHOR]

The November 2024 issue of the American Journal of Physics covers a variety of topics related to physics. Articles include analyzing cat behaviors through classical mechanics, investigating a historic baseball home run, exploring active matter using toy robots, and discussing the Kramers-Kronig relations in optics. Other topics include gravitational lensing, the Sommerfeld atomic model, and the principle of stationary action. Additionally, there are articles on teaching labs for blind students and experimental setups for quantum measurements on single nitrogen-vacancy centers in diamond. [Extracted from the article]

Two-dimensional coupled nonlinear equations of projectile motion with air resistance in the form of quadratic drag are often treated as inseparable and solvable only numerically. However, when they are recast in terms of the angle between the projectile velocity and the horizontal, they become completely uncoupled and possess analytic solutions for projectile velocities as a function of that angle. The equations relating the time and position coordinates to this angle are not integrable in terms of elementary functions but are easy to integrate numerically. Additionally, energy equations explicitly including dissipation terms can be developed as integrals of the equations of motion. One-dimensional numerical integrations can be treated in a pedagogically straightforward way using numerical analysis software or even within a spreadsheet, making this topic accessible to undergraduates. We present this approach with sample numerical results for velocity components, trajectories, and energy-balance of a baseball-sized projectile. [ABSTRACT FROM AUTHOR]

Long-term changes in the tilt of the Earth's axis, relative to the plane of its orbit, are of great significance to long-term climate change, because they control the size of the arctic and Antarctic circles. These "Milankovitch cycles" have hitherto been calculated by classical perturbation methods or by direct numerical integration of Newton's equations of motion. This paper presents an approximate calculation from simple considerations of angular momentum using similar methods to those used to study the precession of a spinning top. It is an instructive exercise in classical mechanics and gives a simple explanation of the phenomenon in terms of angular momentum. It is shown that the main component of "Milankovitch cycles" has a period of 41,000 yr and is due to one of the modes of precession of the Earth-Venus system. The other mode of this system produces a component of period 29,500 yr, and a third component of period 54,000 yr results from the influence of the precession of the orbits of Jupiter and Saturn. These results agree closely with several of the numerical simulations in the literature and strongly suggest that some other results in the literature are incorrect. [ABSTRACT FROM AUTHOR]

We revisit the golfer's curse, which is the possibility that a golf ball can emerge from the cylindrical hole into which it has entered. Our analysis focuses on three constants of the motion. One of these is the energy, because we assume that the ball rolls without slipping on the inner wall of the hole, losing only a small amount of energy to rolling resistance; the other two are related to the angular momentum about the contact point of the ball with the inner wall of the hole. We develop an analysis of the motion of the ball and report measurements of the moment of inertia of a real golf ball. Solving the equation of motion along the vertical direction, we address the question of whether or not the ball could complete a vertical oscillation without reaching the bottom of the hole. We also present measurements of the dynamical friction for a golf ball and discuss dissipation in slip conditions. We conclude by proposing a challenge to golf players: to find a way to send a ball into a hole in order to make it emerge. [ABSTRACT FROM AUTHOR]

We investigate the nonlinear behavior of a simple pendulum fixed to an elastic rod, which can only vibrate horizontally. When released without initial angular momentum, the plane of the pendulum rotates, and the bob traces a delicate stationary pattern. We explain this effect as amplitude modulation due to the nonlinear coupling between the two degrees of freedom. We construct a theoretical model and approximately solve the equations of motion analytically (using the method of multiple scales); we also solve these equations numerically. In the analytical solution, the modulation period depends not only on the dynamical parameters but also on the pendulum's initial release position, which is typical of nonlinear systems. Finally, we build a simple apparatus and conduct a quantitative experiment. The approximate analytical solutions exhibit the same trends as the numerical results and experimental data. After adding linear dissipation, the numerical and experimental results match fairly well. This system can serve as an instructive demonstration as well as a nonlinear dynamics research project for undergraduate students. [ABSTRACT FROM AUTHOR]

We revisit the classical mechanics problem of a particle moving under the influence of a force that depends on its velocity. Using the properties of the rotation matrix and associated operators, we show that it is possible to find an exact analytical solution to a number of problems where the differential equation of motion depends on the velocity. First, we apply our method to the well-known cases of a particle under the influence of the Lorentz force and Coriolis force, providing the complete analytical solution in each case for the motion of the particle in three dimensions. We also show that the complete solution can be obtained when the centrifugal force is included, showing the applicability to cases where there is simultaneous dependence on the position and velocity. This method, which is not currently discussed in a typical course in elementary mechanics, provides an alternative approach to the traditional methods that are used to solve these types of problems. [ABSTRACT FROM AUTHOR]

Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so, one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion and can only depend on the generalized coordinates and velocities, as well as time. In this paper, we establish conditions the Lagrangian and the constraints should obey in order to guarantee that the constraint forces can be obtained consistently. [ABSTRACT FROM AUTHOR]

A ball placed inside a hollow cylinder and spun around an axis perpendicular to the cylinder wall will roll back and forth along the cylinder wall, as expected, but it will also oscillate along the axial direction of the cylinder. These axial oscillations occur in horizontal, tilted, and even vertical cylinders. Here, the equations of motion for this setup are derived and discussed, and experimental results are presented along with a discussion of the observed energy losses. [ABSTRACT FROM AUTHOR]

10.1119/10.0002362.1 Solving the equations of motion for a cylinder with elliptical cross section rolling down an inclined plane constitutes a rather challenging problem. Using Lagrangian mechanics, we have derived the equations of motion analytically and the resulting differential equations are solved numerically. Using a CNC milling machine, we manufactured a solid elliptic cylinder and recorded its dynamics using a high-speed video camera for various inclinations of the plane. We compared theory and experiment by extracting the center of mass coordinates and the angular velocity from the video recording. Limits imposed by the support force reaching zero and insufficient friction to prevent slipping were examined both experimentally and theoretically. [ABSTRACT FROM AUTHOR]

The theory and practical design of a monopole ion trapping system are presented, along with our experimental observations of nonlinear motion of a single charged polyethylene microsphere in the trap. For these observations, the trap was operated at a drive frequency of 70 Hz and easily produced voltages in the kV range. The resulting ion motion was captured using a high-speed camera acquiring images at up to 4000 frames per second. These data yielded time series, phase space portraits, and power spectra of the ion motion in a range of applied voltages, allowing us to observe a period-doubling route to chaos and a period-3 window within the chaotic regime. It is shown that these observations agree with a numerical integration of the ion's nonlinear equation of motion in the monopole electric field. [ABSTRACT FROM AUTHOR]

Two identical objects are simultaneously projected vertically upward with the same initial speed in a uniform gravitational field and then return to their starting point. One object is subject to a resistive force proportional to the nth power of its speed where n ≥ 0 , such as linear (Stokes) drag for n = 1 or quadratic (Newtonian) drag for n = 2. The other object moves through vacuum with no resistance. Which object returns to its starting point first? It is shown analytically that the object subject to drag always wins the race if n ≥ 1 , but that either object can win otherwise, depending on the ratio of the launch speed to the terminal speed of the object subject to drag if 0 < n < 1 or depending on the ratio of the frictional acceleration to the gravitational acceleration if n = 0. These analytical results are confirmed by Euler–Cromer numerical integration of the equations of motion. [ABSTRACT FROM AUTHOR]

With developments in modern instrumentation such as microelectromechanical gyro/accelerometers, high speed video analysis, and precision shaft encoders, there is an increased interest in the study of the large angle oscillations of pendulums as an example of nonlinear dynamics. The solution to the equation of motion for the non-linear pendulum cannot be expressed in terms of elementary functions and is therefore generally approximated by a Fourier series. The present paper extends this to include Fourier coefficients up to A13, as required for amplitudes φ 0 ≈ π. The calculated coefficients are compared with experimental data for a damped and minimally damped pendulum for amplitudes φ 0 ≈ π. [ABSTRACT FROM AUTHOR]

The largest animals are the rorquals, a group of whales which rapidly engulf large aggregations of small-bodied animals along with the water in which they are embedded, with the latter subsequently expulsed via filtration through baleen. Represented by species like the blue, fin, and humpback whales, rorquals can exist in a wide range of body lengths (8–30 m) and masses (4000–190,000 kg). When feeding on krill, kinematic data collected by whale-borne biologging sensors suggest that they first oscillate their flukes several times to accelerate towards their prey, followed by a coasting period with mouth agape as the prey-water mixture is engulfed in a process approximating a perfectly inelastic collision. These kinematic data, used along with momentum conservation and time-averages of a whale's equation of motion, show the largest rorquals as generating significant body forces (10–40 kN) in order to set into forward motion enough engulfed water to at least double overall mass. Interestingly, a scaling analysis of these equations suggests significant reductions in the amount of body force generated per kilogram of body mass at the larger sizes. In other words, and in concert with the allometric growth of the buccal cavity, gigantism would involve smaller fractions of muscle mass to engulf greater volumes of water and prey, thereby imparting a greater efficiency to this unique feeding strategy. [ABSTRACT FROM AUTHOR]

10.1119/10.0001389.1 The azimuthal-radial pendulum is a coupled pendulum-beam system where a pendulum and a supporting string are attached to the free end of a cantilever beam with its other end clamped. Conversion between radial (parallel to cantilever) and azimuthal (perpendicular to cantilever) modes of oscillations is observed when the radial frequency is twice the azimuthal frequency. A theoretical Lagrangian formulation of the system is presented and numerical predictions are compared to experimental data, showing good agreement. A non-linear coupling term arising from the inextensibility of the cantilever beam is shown, via simplifying assumptions that reduce the equations of motion to the Mathieu equation for parametric oscillators, which is responsible for the parametric resonance observed in the system. Experimental work involved measurements of the dynamic trajectory of the pendulum bob in the lab frame, and systematic investigation into the maximum conversion amplitude, beat frequency, and parametric resonance regime boundaries as a function of bob mass, string length, cantilever length, initial release amplitude, and other relevant parameters, shows good agreement with numerical results. [ABSTRACT FROM AUTHOR]

10.1119/10.0000905 A biased ball rolled and spun on a horizontal surface exhibits interesting dynamics. We investigate the motion of a truncated billiard ball, via experiments, analytical methods, and numerical solutions of the equations of motion for a biased sphere rolling without slipping. Solutions are identified where the center of mass moves in a circular or a square path, and we investigate other quasi-periodic motions of the ball. [ABSTRACT FROM AUTHOR]

This paper develops the dynamics and thermodynamics of Stirling engines that run with temperature differences below 100 °C. The working gas pressure is analytically expressed using an alternative thermodynamic cycle. The shaft dynamics is studied using its rotational equation of motion. It is found that the initial volumes of the cold and hot working gas play a non-negligible role in the functioning of the engine. [ABSTRACT FROM AUTHOR]

This paper discusses two approaches for deriving the equations of motion for a ball that rolls without slipping on the surface of a rotating hyperbolic paraboloid. We analyze two possible methods for defining the surface on which the ball rolls, and show the relationship between the two methods. We describe how the stability of the ball's rotation at the saddle point depends on the radius of the ball, in the case where the stability analysis is made in dimensionless parameters. [ABSTRACT FROM AUTHOR]

We describe how orbital tunnels could be used to transport payloads through the Earth. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. Projectiles move along an orbital tunnel faster than they would along a brachistochrone connecting the same points but the body force components cancel. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. We use models of the interior of the Earth with three values of the polytropic index (n) to calculate interior orbits that travel between surface points. The n = 3 results are also scaled to the Sun. Numerical integrations of the equations describing polytropes were used to generate the initial models. Numerical integration of the equations of motion are then used to calculate the angular distance you can travel along the surface and the traversal time as a function of the parabolic periapsis distance for each model. Trajectories through objects of low central condensation show a focussing effect that decreases as the central condensation increases. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. The results can be scaled to other planets, stars, or even globular clusters. [ABSTRACT FROM AUTHOR]

We study the nonlinear behaviors of mass-spring systems damped by dry friction using simulation by a nonlinear LC circuit damped by anti-parallel diodes. We show that the differential equation for the electric oscillator is equivalent to that of the mechanical system when a piecewise linear model is used to simplify the diodes' I–V curve. We derive series solutions to the differential equation under weak nonlinear approximation which can describe the resonant response as well as amplitudes of superharmonic components. The experimental results are consistent with the series solutions. We also present the phenomenon of hysteresis. A theoretical analysis along with numerical simulations is conducted to explore the stick-slip boundary. The correspondence between the mechanical and electric oscillators makes it easy to demonstrate the behaviors of this nonlinear oscillator on a digital oscilloscope. It can be used to extend the linear RLC experiment at the undergraduate level. [ABSTRACT FROM AUTHOR]

A Bussard ramjet is a relativistic spacecraft, fueled by fusion energy of cosmic matter that is collected during the flight. We derive the equation of motion of such a spaceship for a given mass density in space and the fusion mass defect. Two ramjet engine scenarios, where the thrust for propulsion is generated by emission of photons or acceleration of matter, are outlined. As long as not all collected matter is transformed into fusion energy, mass engines are superior to photon engines. If the collected matter is stopped by the spacecraft before fusion it may not reach relativistic terminal velocities. For an ideal ramjet, where no matter is stopped for the generation of energy for propulsion, endless acceleration and relativistic velocities may be obtained such that crossing the universe in a human lifespan would be possible. A journey along one space coordinate and the smallest possible radii of curves were evaluated. The results are compared to the plots in the novel “Tau Zero” by Poul Anderson. [ABSTRACT FROM AUTHOR]

We investigate the development of student understanding of Newton's laws using a pre-instruction test (the Force Concept Inventory), followed by a series of post-instruction tests and interviews. While some students' somewhat naive, pre-existing models of Newton's third law are largely eliminated following a semester of teaching, we find that a particular inconsistent model is highly resilient to, and may even be strengthened by, instruction. If test items contain words that cue students to think of Newton's second law, then students are more likely to apply a "net force" approach to solving problems, even if it is inappropriate to do so. Additional instruction, reinforcing physical concepts in multiple settings and from multiple sources, appears to help students develop a more connected and consistent level of understanding. We recommend explicitly encouraging students to check their work for consistency with physical principles, along with the standard checks for dimensionality and order of magnitude, to encourage reflective and rigorous problem solving. [ABSTRACT FROM AUTHOR]

Noether's theorem is widely regarded as one of the most elegant results in theoretical physics. The article presents two simple examples that can be used to demonstrate the basic idea behind Noether's theorem, by deriving a relation between translational symmetry in the time coordinate and energy conservation in a way that is accessible to beginning students. [ABSTRACT FROM AUTHOR]

Drone-drone collisions over New York City via the kinetic theory of gases I Edward W. Walbridge i B 89 b (10), p. 930 https://doi.org/10.1119/10.0005035 If the predictions made by companies, such as Amazon, hold true, more than 50% of parcel deliveries will be made by drone aircraft within the present decade. In this article, the author uses kinetic theory to estimate the surprisingly high collision rate between hypothetical "dumb" drones, which have no collision avoidance capability, and discusses the required reliability of such systems to ensure safe operation. These brief summaries are designed to help readers easily see which articles will be most valuable to them. [Extracted from the article]

We examine the motions of particles in quadrupole ion traps as a function of damping and trapping forces, including cases where nonlinear damping or nonlinearities in the electric field geometry play significant roles. In the absence of nonlinearities, particles are either damped to the trap center or ejected, while their addition brings about a rich spectrum of stable closed particle trajectories. In three-dimensional (3D) quadrupole traps, the extended orbits are typically confined to the trap axis, and for this case we present a 1D analysis of the relevant equation of motion. We follow this with an analysis of 2D quadrupole traps that frequently show diamond-shaped closed orbits. For both the 1D and 2D cases, we present experimental observations of the calculated trajectories in microparticle ion traps. We also report the discovery of a new collective behavior in damped 2D microparticle ion traps, where particles spontaneously assemble into a remarkable knot of overlapping, corotating diamond orbits, self-stabilized by air currents arising from the particle motion. [ABSTRACT FROM AUTHOR]

We derive the rotational form of Newton's second law τ = Iα from the translational form ... = m... by performing a force analysis of a simple body consisting of two discrete masses. Curiously, a truly rigid body model leads to an incorrect statement of the rotational second law. The failure of this model is traced to its violation of the strong form of Newton's third law. This leads us to consider a slightly modified non-rigid model that respects the third law, produces the correct rotational second law, and makes explicit the importance of the product of the tangential force with the radial distance: the torque. [ABSTRACT FROM AUTHOR]

The motion of a point particle sliding on a turntable is studied. The equations of motion are derived assuming that the table exerts a frictional force on the particle that is of constant magnitude and directed opposite to the particle's motion relative to the turntable. After expressing the equations in terms of dimensionless variables, some of the general properties of the solutions are discussed. Approximate analytic solutions are found for the cases in which (i) the particle is released from rest with respect to the lab frame, and (ii) the particle is released from rest with respect to the turntable. The equations are then solved numerically to get a more complete understanding of the motion. It is found that one can define an escape speed for the particle, which is the minimum speed required to get the particle to move out to infinity. The escape speed is a function of both the distance from the center of the turntable and the direction of the initial velocity. A qualitative explanation of this behavior is given in terms of fictitious forces. Numerical study also indicates an alternative way to measure the coefficient of friction between the particle and the turntable. [ABSTRACT FROM AUTHOR]

The recent discovery of thirteen new and distinct three-body periodic planar orbits suggests that many more such orbits remain undiscovered. Searches in two-dimensional subspaces of the full four-dimensional space of initial conditions require computing resources that are available to many students, and the required level of computational and numerical expertise is also at the advanced undergraduate level. We discuss the methods for solving the planar three-body equations of motion, as well as some basic strategies and tactics for searches of periodic orbits. Our discussion should allow interested undergraduates to start their own searches. Users can submit new three-body orbits to a wiki-based website. [ABSTRACT FROM AUTHOR]

Schrödinger's equation as a diffusion equation I Katsunori Mita i B 89 b (5), p. 500 https://doi.org/10.1119/10.0002765 Schrödinger's equation is more often than not considered to be a wave equation for the wave function. Actually, they only agreed on the assumption that particle velocity can be equaled to the wave group velocity. You will get a new perspective on classic undergraduate quantum mechanics problems: a particle in a box, Gaussian wave packets, the harmonic oscillator, and a mind-boggling quantum mechanical soliton. [Extracted from the article]

We present a toy model of anaerobic glycolysis that utilizes appropriate physiological and mathematical consideration while remaining useful to the athlete. The toy model produces an optimal pacing strategy for 400-m and 800-m races that is analytically calculated via the Euler-Lagrange equation. The calculation of the optimum v(t) is presented in detail, with an emphasis on intuitive arguments in order to serve as a bridge between the basic techniques presented in undergraduate physics textbooks and the more advanced techniques of control theory. Observed pacing strategies in 400-m and 800-m world-record races are found to be well-fit by the toy model, which allows us to draw a new physiological interpretation for the advantages of common weight-training practices. [ABSTRACT FROM AUTHOR]

The equation of motion for rolling bodies can be derived in several different ways. This paper shows that the equation of motion can be derived in five and only five ways without needing to know the normal and frictional forces acting at the point of contact. After enumerating these five ways, we examine two illustrative examples: an asymmetric disk rolling on a cylinder and a symmetric ball rolling on a turntable. We also discuss the educational benefits of including this topic when teaching mechanics. [ABSTRACT FROM AUTHOR]

An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ħ→ 0. Several ways in which this limit may be performed are considered. It is shown that Schrrdinger's equation for a single particle moving in an external potential V does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics--at least in this sense--upon straightforward application of the limit ħ → 0. [ABSTRACT FROM AUTHOR]

The idea of predicting the future from the knowledge of the past is quite natural, even when dealing with systems whose equations of motion are not known. This long-standing issue is revisited in the light of modern ergodic theory of dynamical systems and becomes particularly interesting from a pedagogical perspective due to its close link with Poincaré's recurrence. Using such a connection, a very general result of ergodic theory--Kac's lemma--can be used to establish the intrinsic limitations to the possibility of predicting the future from the past. In spite of a naive expectation, predictability is hindered more by the effective number of degrees of freedom of a system than by the presence of chaos. If the effective number of degrees of freedom becomes large enough, whether the system is chaotic or not, predictions turn out to be practically impossible. The discussion of these issues is illustrated with the help of the numerical study of simple models. [ABSTRACT FROM AUTHOR]

We consider classical Yang-Mills theory with point sources and derive equations of motion for the coupled particle-field system. As an example, we discuss the special case of Yang-Mills theory coupled to point particles in (1 + 1) dimensions. We formulate the initial value problem for the system and describe several example solutions. [ABSTRACT FROM AUTHOR]

While the d'Alembert-Lagrange principle has been widely used to derive equations of state for dynamical systems under holonomic (geometric) and non-integrable linear-velocity (kinematic) constraints, its application to general kinematic constraints with a general velocity and acceleration-dependence has remained elusive, mainly because there is no clear method, whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. We show how this limitation can be resolved by requiring that the states displaced by the variation are compatible with the kinematic constraints. A set of linear auxiliary conditions on the displacements is established and adjoined to the d'Alembert-Lagrange equation via Lagrange's multipliers to yield the equations of state. As a consequence, new transpositional relations satisfied by the velocity and acceleration displacements are also established. The theory is tested for a quadratic velocity constraint and for a nonholonomic penny rolling and turning upright on an inclined plane. [ABSTRACT FROM AUTHOR]

It is shown that energy and momentum are conserved during the runaway motion of a radiating charge and during free fall of a charge in a gravitational field. The runaway motion demonstrates the consistency of classical electrodynamics and the Lorentz-Abraham-Dirac equation. The important role of the Schott (acceleration) energy in this connection is made clear, and it is shown that the Schott energy is the part of the electromagnetic field energy that is proportional to (minus) the scalar product of the velocity and acceleration of a moving accelerated charged particle. The Schott energy is negative if the acceleration is in the same direction as the velocity and positive if it is opposite. During runaway motion, the Schott energy becomes more and more negative, and for a charged particle, it is localized at the particle. It is also shown that a proton and a neutron fall with the same acceleration in a uniform gravitational field, although the proton radiates and the neutron does not. The radiation energy comes from the Schott energy. [ABSTRACT FROM AUTHOR]

This paper considers a model of free fall in the troposphere (altitude of up to II km) that takes into account the decrease of the air density, temperature, and the speed of sound. Two cases are analyzed: The maximum speed is less than Mach 0.8 (subsonic fall) for which the drag coefficient is constant, and the maximum speed exceeds Mach 0.8 (transonic and supersonic fall) for which the drag coefficient depends on the speed. The equation of motion is solved analytically for subsonic fall, and the range of validity of the solution is found. The solution for supersonic fall in the troposphere is found for the free fall of a bomb. The drag coefficient function is obtained by a least square fit to measured data, and the differential equation is solved numerically. [ABSTRACT FROM AUTHOR]