The physics of a bouncing ball concerns the physical behaviour of bouncing balls, particularly its motion before, during, and after impact against the surface of another body. Several aspects of a bouncing ball's behaviour serve as an introduction to mechanics in high school or undergraduate level physics courses. However, the exact modelling of the behaviour is complex and of interest in sports engineering.
The motion of a ball is generally described by projectile motion (which can be affected by gravity, drag, the Magnus effect, and buoyancy), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure). To ensure fair play, many sports governing bodies set limits on the bounciness of their ball and forbid tampering with the ball's aerodynamic properties. The bounciness of balls has been a feature of sports as ancient as the Mesoamerican ballgame. [1]
The motion of a bouncing ball obeys projectile motion. [2] [3] Many forces act on a real ball, namely the gravitational force (FG), the drag force due to air resistance (FD), the Magnus force due to the ball's spin (FM), and the buoyant force (FB). In general, one has to use Newton's second law taking all forces into account to analyze the ball's motion:
where m is the ball's mass. Here, a, v, r represent the ball's acceleration, velocity, and position over time t.
The gravitational force is directed downwards and is equal to [4]
where m is the mass of the ball, and g is the gravitational acceleration, which on Earth varies between 9.764 m/s2 and 9.834 m/s2. [5] Because the other forces are usually small, the motion is often idealized as being only under the influence of gravity. If only the force of gravity acts on the ball, the mechanical energy will be conserved during its flight. In this idealized case, the equations of motion are given by
where a, v, and r denote the acceleration, velocity, and position of the ball, and v0 and r0 are the initial velocity and position of the ball, respectively.
More specifically, if the ball is bounced at an angle θ with the ground, the motion in the x- and y-axes (representing horizontal and vertical motion, respectively) is described by [6]
x-axis | y-axis |
---|---|
The equations imply that the maximum height (H) and range (R) and time of flight (T) of a ball bouncing on a flat surface are given by [2] [6]
Further refinements to the motion of the ball can be made by taking into account air resistance (and related effects such as drag and wind), the Magnus effect, and buoyancy. Because lighter balls accelerate more readily, their motion tends to be affected more by such forces.
Air flow around the ball can be either laminar or turbulent depending on the Reynolds number (Re), defined as:
where ρ is the density of air, μ the dynamic viscosity of air, D the diameter of the ball, and v the velocity of the ball through air. At a temperature of 20 °C, ρ = 1.2 kg/m3 and μ = 1.8×10−5 Pa·s. [7]
If the Reynolds number is very low (Re < 1), the drag force on the ball is described by Stokes' law: [8]
where r is the radius of the ball. This force acts in opposition to the ball's direction (in the direction of ). For most sports balls, however, the Reynolds number will be between 104 and 105 and Stokes' law does not apply. [9] At these higher values of the Reynolds number, the drag force on the ball is instead described by the drag equation: [10]
where Cd is the drag coefficient, and A the cross-sectional area of the ball.
Drag will cause the ball to lose mechanical energy during its flight, and will reduce its range and height, while crosswinds will deflect it from its original path. Both effects have to be taken into account by players in sports such as golf.
The spin of the ball will affect its trajectory through the Magnus effect. According to the Kutta–Joukowski theorem, for a spinning sphere with an inviscid flow of air, the Magnus force is equal to [11]
where r is the radius of the ball, ω the angular velocity (or spin rate) of the ball, ρ the density of air, and v the velocity of the ball relative to air. This force is directed perpendicular to the motion and perpendicular to the axis of rotation (in the direction of ). The force is directed upwards for backspin and downwards for topspin. In reality, flow is never inviscid, and the Magnus lift is better described by [12]
where ρ is the density of air, CL the lift coefficient, A the cross-sectional area of the ball, and v the velocity of the ball relative to air. The lift coefficient is a complex factor which depends amongst other things on the ratio rω/v, the Reynolds number, and surface roughness. [12] In certain conditions, the lift coefficient can even be negative, changing the direction of the Magnus force (reverse Magnus effect). [4] [13] [14]
In sports like tennis or volleyball, the player can use the Magnus effect to control the ball's trajectory (e.g. via topspin or backspin) during flight. In golf, the effect is responsible for slicing and hooking which are usually a detriment to the golfer, but also helps with increasing the range of a drive and other shots. [15] [16] In baseball, pitchers use the effect to create curveballs and other special pitches. [17]
Ball tampering is often illegal, and is often at the centre of cricket controversies such as the one between England and Pakistan in August 2006. [18] In baseball, the term 'spitball' refers to the illegal coating of the ball with spit or other substances to alter the aerodynamics of the ball. [19]
Any object immersed in a fluid such as water or air will experience an upwards buoyancy. [20] According to Archimedes' principle, this buoyant force is equal to the weight of the fluid displaced by the object. In the case of a sphere, this force is equal to
The buoyant force is usually small compared to the drag and Magnus forces and can often be neglected. However, in the case of a basketball, the buoyant force can amount to about 1.5% of the ball's weight. [20] Since buoyancy is directed upwards, it will act to increase the range and height of the ball.
External videos | |
---|---|
Florian Korn (2013). "Ball bouncing in slow motion: Rubber ball". YouTube. |
When a ball impacts a surface, the surface recoils and vibrates, as does the ball, creating both sound and heat, and the ball loses kinetic energy. Additionally, the impact can impart some rotation to the ball, transferring some of its translational kinetic energy into rotational kinetic energy. This energy loss is usually characterized (indirectly) through the coefficient of restitution (or COR, denoted e): [23] [note 1]
where vf and vi are the final and initial velocities of the ball, and uf and ui are the final and initial velocities impacting surface, respectively. In the specific case where a ball impacts on an immovable surface, the COR simplifies to
For a ball dropped against a floor, the COR will therefore vary between 0 (no bounce, total loss of energy) and 1 (perfectly bouncy, no energy loss). A COR value below 0 or above 1 is theoretically possible, but would indicate that the ball went through the surface (e < 0), or that the surface was not "relaxed" when the ball impacted it (e > 1), like in the case of a ball landing on spring-loaded platform.
To analyze the vertical and horizontal components of the motion, the COR is sometimes split up into a normal COR (ey), and tangential COR (ex), defined as [24]
where r and ω denote the radius and angular velocity of the ball, while R and Ω denote the radius and angular velocity the impacting surface (such as a baseball bat). In particular rω is the tangential velocity of the ball's surface, while RΩ is the tangential velocity of the impacting surface. These are especially of interest when the ball impacts the surface at an oblique angle, or when rotation is involved.
For a straight drop on the ground with no rotation, with only the force of gravity acting on the ball, the COR can be related to several other quantities by: [22] [25]
Here, K and U denote the kinetic and potential energy of the ball, H is the maximum height of the ball, and T is the time of flight of the ball. The 'i' and 'f' subscript refer to the initial (before impact) and final (after impact) states of the ball. Likewise, the energy loss at impact can be related to the COR by
The COR of a ball can be affected by several things, mainly
External conditions such as temperature can change the properties of the impacting surface or of the ball, making them either more flexible or more rigid. This will, in turn, affect the COR. [22] In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR. [22] [28]
External videos | |
---|---|
BiomechanicsMMU (2008). "Golf impacts - Slow motion video". YouTube. |
Upon impacting the ground, some translational kinetic energy can be converted to rotational kinetic energy and vice versa depending on the ball's impact angle and angular velocity. If the ball moves horizontally at impact, friction will have a 'translational' component in the direction opposite to the ball's motion. In the figure, the ball is moving to the right, and thus it will have a translational component of friction pushing the ball to the left. Additionally, if the ball is spinning at impact, friction will have a 'rotational' component in the direction opposite to the ball's rotation. On the figure, the ball is spinning clockwise, and the point impacting the ground is moving to the left with respect to the ball's center of mass. The rotational component of friction is therefore pushing the ball to the right. Unlike the normal force and the force of gravity, these frictional forces will exert a torque on the ball, and change its angular velocity (ω). [29] [30] [31] [32]
Three situations can arise: [32] [33] [34]
If the surface is inclined by some amount θ, the entire diagram would be rotated by θ, but the force of gravity would remain pointing downwards (forming an angle θ with the surface). Gravity would then have a component parallel to the surface, which would contribute to friction, and thus contribute to rotation. [32]
In racquet sports such as table tennis or racquetball, skilled players will use spin (including sidespin) to suddenly alter the ball's direction when it impacts surface, such as the ground or their opponent's racquet. Similarly, in cricket, there are various methods of spin bowling that can make the ball deviate significantly off the pitch.
The bounce of an oval-shaped ball (such as those used in gridiron football or rugby football) is in general much less predictable than the bounce of a spherical ball. Depending on the ball's alignment at impact, the normal force can act ahead or behind the centre of mass of the ball, and friction from the ground will depend on the alignment of the ball, as well as its rotation, spin, and impact velocity. Where the forces act with respect to the centre of mass of the ball changes as the ball rolls on the ground, and all forces can exert a torque on the ball, including the normal force and the force of gravity. This can cause the ball to bounce forward, bounce back, or sideways. Because it is possible to transfer some rotational kinetic energy into translational kinetic energy, it is even possible for the COR to be greater than 1, or for the forward velocity of the ball to increase upon impact. [35]
External videos | |
---|---|
Physics Girl (2015). "Stacked Ball Drop". YouTube. |
A popular demonstration involves the bounce of multiple stacked balls. If a tennis ball is stacked on top of a basketball, and the two of them are dropped at the same time, the tennis ball will bounce much higher than it would have if dropped on its own, even exceeding its original release height. [36] [37] The result is surprising as it apparently violates conservation of energy. [38] However, upon closer inspection, the basketball does not bounce as high as it would have if the tennis ball had not been on top of it, and transferred some of its energy into the tennis ball, propelling it to a greater height. [36]
The usual explanation involves considering two separate impacts: the basketball impacting with the floor, and then the basketball impacting with the tennis ball. [36] [37] Assuming perfectly elastic collisions, the basketball impacting the floor at 1 m/s would rebound at 1 m/s. The tennis ball going at 1 m/s would then have a relative impact velocity of 2 m/s, which means it would rebound at 2 m/s relative to the basketball, or 3 m/s relative to the floor, and triple its rebound velocity compared to impacting the floor on its own. This implies that the ball would bounce to 9 times its original height. [note 2] In reality, due to inelastic collisions, the tennis ball will increase its velocity and rebound height by a smaller factor, but still will bounce faster and higher than it would have on its own. [37]
While the assumptions of separate impacts is not actually valid (the balls remain in close contact with each other during most of the impact), this model will nonetheless reproduce experimental results with good agreement, [37] and is often used to understand more complex phenomena such as the core collapse of supernovae, [36] or gravitational slingshot manoeuvres. [39]
Several sports governing bodies regulate the bounciness of a ball through various ways, some direct, some indirect.
The pressure of an American football was at the center of the deflategate controversy. [50] [51] Some sports do not regulate the bouncing properties of balls directly, but instead specify a construction method. In baseball, the introduction of a cork-based ball helped to end the dead-ball era and trigger the live-ball era. [52] [53]
Consider the following potential a bouncing ball is subjected to:
The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential . The classical turning points are identified and . Thus applying the quantization condition obtained in WKB:
Letting where , solving for with given , we get the quantum mechanical energy of a bouncing ball: [54]
In physics, angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.
In physics, specifically in electromagnetism, the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of
In Newtonian mechanics, momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity, then the object's momentum p is:
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.
Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to north pole of the magnet.
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
Euler's Disk, invented between 1987 and 1990 by Joseph Bendik, is a trademarked scientific educational toy. It is used to illustrate and study the dynamic system of a spinning and rolling disk on a flat or curved surface. It has been the subject of several scientific papers.
In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is at the origin of magnetocrystalline anisotropy and the spin Hall effect.
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector.
Rotation around a fixed axis or axial rotation is a special case of rotational motion around an axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.
The coefficient of restitution, is the ratio of the relative velocity of separation after collision to the relative velocity of approach before collision. It can aIso be defined as the square root of the ratio of the final kinetic energy to the initial kinetic energy. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested.
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.
Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.