Bouncing ball dynamics
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The physics of a bouncing ball concerns the physical behaviour of bouncing
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
s, particularly its
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
before, during, and after
impact Impact may refer to: * Impact (mechanics), a high force or shock (mechanics) over a short time period * Impact, Texas, a town in Taylor County, Texas, US Science and technology * Impact crater, a meteor crater caused by an impact event * Impact ...
against the surface of another
body Body may refer to: In science * Physical body, an object in physics that represents a large amount, has mass or takes up space * Body (biology), the physical material of an organism * Body plan, the physical features shared by a group of anima ...
. Several aspects of a bouncing ball's behaviour serve as an introduction to
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
in
high school A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' lower secondary education'' (ages 11 to 14) and ''upper seconda ...
or
undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-lev ...
level physics courses. However, the exact modelling of the behaviour is complex and of interest in
sports engineering Aerobics engineering is a fast growing field of engineering which encompasses the design and production of sports equipment and facilities, performance measurement and athletic feedback systems, and the study of kinematics, dynamics and biomec ...
. The motion of a ball is generally described by projectile motion (which can be affected by
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, drag, the Magnus effect, and
buoyancy Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
), while its impact is usually characterized through the
coefficient of restitution The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectl ...
(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 Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
). To ensure
fair play Fair play or Fairplay usually refers to sportsmanship. Fair play or Fairplay may also refer to: Media * ''Fair Play'' (1925 film), an American silent film * ''Fair Play'', a 1972 TV movie starring Paul Ford * ''Fair Play'' (2014 film), a Czech ...
, many
sports governing bodies A sports governing body is a sports organization that has a regulatory or sanctioning function. Sports governing bodies come in various forms and have a variety of regulatory functions. Examples of this can include disciplinary action for rule ...
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.


Forces during flight and effect on motion

The motion of a bouncing ball obeys projectile motion. Many forces act on a real ball, namely the gravitational force (FG), the
drag force In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
due to air resistance (FD), the
Magnus force The Magnus effect is an observable phenomenon commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner not present when the object is not spinning. The deflection can be expl ...
due to the ball's
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
(FM), and the
buoyant force Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
(FB). In general, one has to use
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
taking all forces into account to analyze the ball's motion: :\begin \sum \mathbf & = m\mathbf, \\ \mathbf_\text + \mathbf_\text + \mathbf_\text + \mathbf_\text & = m \mathbf = m \frac = m\frac, \end where ''m'' is the ball's mass. Here, a, v, r represent the ball's
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
,
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, and
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
over
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
''t''.


Gravity

The gravitational force is directed downwards and is equal to :F_\text = mg, where ''m'' is the mass of the ball, and ''g'' is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
, which on
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
varies between and . 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 :\begin \mathbf & = -g \mathbf, \\ \mathbf & = \mathbf_\text + \mathbft, \\ \mathbf & = \mathbf_0 + \mathbf_0 t + \frac\mathbft^2, \end 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 The equations imply that the maximum height (''H'') and
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
(''R'') and
time of flight Time of flight (ToF) is the measurement of the time taken by an object, particle or wave (be it acoustic, electromagnetic, etc.) to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a w ...
(''T'') of a ball bouncing on a flat surface are given by :\begin H & = \frac\sin^2\left(\theta\right), \\ R &= \frac\sin\left(2\theta\right),~\text \\ T &= \frac \sin \left(\theta \right). \end 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 Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hou ...
), the Magnus effect, and
buoyancy Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
. Because lighter balls accelerate more readily, their motion tends to be affected more by such forces.


Drag

Air flow around the ball can be either
laminar Laminar means "flat". Laminar may refer to: Terms in science and engineering: * Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules * Laminar armour or "band ...
or
turbulent In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
depending on the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
(Re), defined as: :\text = \frac, where ''ρ'' is the density of air, ''μ'' the
dynamic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
of air, ''D'' the diameter of the ball, and ''v'' the velocity of the ball through air. At a
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of , and . If the Reynolds number is very low (Re < 1), the drag force on the ball is described by
Stokes' law In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by s ...
: :F_\text = 6 \pi \mu r v, where ''r'' is the radius of the ball. This force acts in opposition to the ball's direction (in the direction of \textstyle -\hat \mathbf). For most sports balls, however, the Reynolds number will be between 104 and 105 and Stokes' law does not apply. At these higher values of the Reynolds number, the drag force on the ball is instead described by the drag equation: :F_\text = \frac \rho C_\text A v^2, where ''C''d is the drag coefficient, and ''A'' the
cross-sectional area In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
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.


Magnus effect

The
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
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 In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as ...
of air, the Magnus force is equal to :F_\text = \frac \pi r^3 \rho \omega v, where ''r'' is the radius of the ball, ''ω'' the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
(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 \textstyle \hat \mathbf \times \hat \mathbf). 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 :F_\text=\frac\rho C_\text A v^2, where ''ρ'' is the density of air, ''C''L 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 Surface roughness, often shortened to roughness, is a component of surface finish (surface texture). It is quantified by the deviations in the direction of the normal vector of a real surface from its ideal form. If these deviations are large, ...
. In certain conditions, the lift coefficient can even be negative, changing the direction of the Magnus force (
reverse Magnus effect Reverse or reversing may refer to: Arts and media * ''Reverse'' (Eldritch album), 2001 * ''Reverse'' (2009 film), a Polish comedy-drama film * ''Reverse'' (2019 film), an Iranian crime-drama film * ''Reverse'' (Morandi album), 2005 * ''Reverse'' ...
). In sports like
tennis Tennis is a racket sport that is played either individually against a single opponent ( singles) or between two teams of two players each ( doubles). Each player uses a tennis racket that is strung with cord to strike a hollow rubber ball ...
or
volleyball Volleyball is a team sport in which two teams of six players are separated by a net. Each team tries to score points by grounding a ball on the other team's court under organized rules. It has been a part of the official program of the Summ ...
, the player can use the Magnus effect to control the ball's trajectory (e.g. via topspin or backspin) during flight. In
golf Golf is a club-and-ball sport in which players use various clubs to hit balls into a series of holes on a course in as few strokes as possible. Golf, unlike most ball games, cannot and does not use a standardized playing area, and coping wi ...
, 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 Drive or The Drive may refer to: Motoring * Driving, the act of controlling a vehicle * Road trip, a journey on roads Roadways Roadways called "drives" may include: * Driveway, a private road for local access to structures, abbreviated "drive" ...
and other shots. In
baseball Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding tea ...
,
pitcher In baseball, the pitcher is the player who throws ("pitches") the baseball from the pitcher's mound toward the catcher to begin each play, with the goal of retiring a batter, who attempts to either make contact with the pitched ball or draw ...
s use the effect to create
curveball In baseball and softball, the curveball is a type of pitch thrown with a characteristic grip and hand movement that imparts forward spin to the ball, causing it to dive as it approaches the plate. Varieties of curveball include the 12–6 curve ...
s and other special pitches.
Ball tampering In the sport of cricket, ball tampering is an action in which a fielder illegally alters the condition of the ball. The primary motivation of ball tampering is to interfere with the aerodynamics of the ball to aid swing bowling. Definition Unde ...
is often illegal, and is often at the centre of
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by striki ...
controversies such as the one between England and Pakistan in August 2006. In baseball, the term ' spitball' refers to the illegal coating of the ball with spit or other substances to alter the
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
of the ball.


Buoyancy

Any object immersed in a
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
such as water or air will experience an upwards
buoyancy Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...
. 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 :F_\text = \frac\pi r^3 \rho g. 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. Since buoyancy is directed upwards, it will act to increase the range and height of the ball.


Impact

When a ball impacts a surface, the surface recoils and vibrates, as does the ball, creating both
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
and
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
, and the ball loses
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
. Additionally, the impact can impart some rotation to the ball, transferring some of its
translational kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
into
rotational kinetic energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ...
. This energy loss is usually characterized (indirectly) through the
coefficient of restitution The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectl ...
(or COR, denoted ''e''):Here, ''v'' and ''u'' are not just the magnitude of velocities, but include also their direction (
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
).
:e = -\frac, where ''v''f and ''v''i are the final and initial velocities of the ball, and ''u''f and ''u''i 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 :e = -\frac. 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 (), or that the surface was not "relaxed" when the ball impacted it (), 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 (''e''y), and '' tangential'' COR (''e''x), defined as :e_\text = -\frac, :e_\text = -\frac, 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 In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity. ...
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 In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are ...
, or when
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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: :e = \left, \frac\ = \sqrt = \sqrt = \sqrt = \frac =\sqrt. Here, ''K'' and ''U'' denote the
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
and
potential energy 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. Common types of potential energy include the gravitational potentia ...
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 :\text = \frac \times 100\% = \left( 1 - e^2 \right) \times 100\%. The COR of a ball can be affected by several things, mainly * the nature of the impacting surface (e.g. grass, concrete, wire mesh) * the material of the ball (e.g. leather, rubber, plastic) * the pressure inside the ball (if hollow) * the amount of rotation induced in the ball at impact * the impact velocity External conditions such as
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
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. In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR.


Spin and angle of impact

Upon impacting the ground, some
translational kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
can be converted to
rotational kinetic energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the ...
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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. 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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
on the ball, and change its angular velocity (''ω''). Three situations can arise: #If a ball is propelled forward with '' backspin'', the translational and rotational friction will act in the same directions. The ball's angular velocity will be reduced after impact, as will its horizontal velocity, and the ball is propelled ''upwards'', possibly even exceeding its original height. It is also possible for the ball to start spinning in the opposite direction, and even bounce backwards. #If a ball is propelled forward with '' topspin'', the translational and rotational friction act will act in opposite directions. What exactly happens depends on which of the two components dominate. ## If the ball is spinning much more rapidly than it was moving, rotational friction will dominate. The ball's angular velocity will be reduced after impact, but its horizontal velocity will be increased. The ball will be propelled ''forward'' but will not exceed its original height, and will keep spinning in the same direction. ## If the ball is moving much more rapidly than it was spinning, translational friction will dominate. The ball's angular velocity will be increased after impact, but its horizontal velocity will be decreased. The ball will not exceed its original height and will keep spinning in the same direction. 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. In
racquet sports Racket sports are games in which players use a racket or paddle to hit a ball or other object. Rackets consist of a handled frame with an open hoop that supports a network of tightly stretched strings. Paddles have a solid face rather than a ne ...
such as
table tennis Table tennis, also known as ping-pong and whiff-whaff, is a sport in which two or four players hit a lightweight ball, also known as the ping-pong ball, back and forth across a table using small solid rackets. It takes place on a hard table div ...
or racquetball, skilled players will use spin (including
sidespin This glossary provides definitions and context for terminology related to, and jargon specific to, the sport of pickleball. Words or phrases in italics can be found on the list in their respective alphabetic sections. 0–9 ;0–0:See '' Zero ...
) to suddenly alter the ball's direction when it impacts surface, such as the ground or their opponent's racquet. Similarly, in
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by striki ...
, there are various methods of spin bowling that can make the ball deviate significantly off the pitch.


Non-spherical balls

The bounce of an oval-shaped ball (such as those used in
gridiron football Gridiron football,"Gridiron football"
''Encyclopædia Britannica''. Ret ...
or
rugby football Rugby football is the collective name for the team sports of rugby union and rugby league. Canadian football and, to a lesser extent, American football were once considered forms of rugby football, but are seldom now referred to as such. The ...
) 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
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.


Multiple stacked balls

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. The result is surprising as it apparently violates conservation of energy. 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. The usual explanation involves considering two separate impacts: the basketball impacting with the floor, and then the basketball impacting with the tennis ball. 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.Since conservation of mechanical energy implies \textstyle \fracmv_\text^2 = mgH_\text, then \textstyle H_\text is proportional to v^2_\text. In reality, due to
inelastic collision An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energ ...
s, 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. 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, and is often used to understand more complex phenomena such as the core collapse of
supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or when ...
e, or gravitational slingshot manoeuvres.


Sport regulations

Several
sports governing bodies A sports governing body is a sports organization that has a regulatory or sanctioning function. Sports governing bodies come in various forms and have a variety of regulatory functions. Examples of this can include disciplinary action for rule ...
regulate the bounciness of a ball through various ways, some direct, some indirect. *
AFL AFL may refer to: Sports * American Football League (AFL), a name shared by several separate and unrelated professional American football leagues: ** American Football League (1926) (a.k.a. "AFL I"), first rival of the National Football Leagu ...
: Regulates the gauge pressure of the
football Football is a family of team sports that involve, to varying degrees, kicking a ball to score a goal. Unqualified, the word ''football'' normally means the form of football that is the most popular where the word is used. Sports commonly c ...
to be between and . *
FIBA The International Basketball Federation (FIBA ; French: ) is an association of national organizations which governs the sport of basketball worldwide. Originally known as the (hence FIBA), in 1989 it dropped the word ''amateur'' from its na ...
: Regulates the gauge pressure so the
basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular Basketball court, court, compete with the primary objective of #Shooting, shooting a basketball (ball), basketball (appr ...
bounces between 1200 mm and 1400 mm (top of the ball) when it is dropped from a height of 1800 mm (bottom of the ball). This roughly corresponds to a COR of 0.727 to 0.806.Calculated using \textstyle e = \sqrt and (if applicable) the diameter of the ball. The calculation assumes air resistance is negligible. *
FIFA FIFA (; stands for ''Fédération Internationale de Football Association'' ( French), meaning International Association Football Federation ) is the international governing body of association football, beach football and futsal. It was found ...
: Regulates the gauge pressure of the soccer ball to be between of and at
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
(61 to 111 
kPa KPA may refer to: * Keele Postgraduate Association, Keele University, UK, formerly Keele Research Association (KRA) * Kensington (Olympia) station, London, England, National Rail station code * Kenya Ports Authority * ''Kiln phosphoric acid'', a d ...
). * FIVB: Regulates the gauge pressure of the
volleyball Volleyball is a team sport in which two teams of six players are separated by a net. Each team tries to score points by grounding a ball on the other team's court under organized rules. It has been a part of the official program of the Summ ...
to be between to (29.4 to 31.9 kPa) for
indoor volleyball Indoor(s) may refer to: *the interior of a building *Indoor environment, in building science, traditionally includes the study of indoor thermal environment, indoor acoustic environment, indoor light environment, and indoor air quality *Built envi ...
, and to (17.2 to 22.1 kPa) for
beach volleyball Beach volleyball is a team sport played by two teams of two or more players on a sand court divided by a net. Similar to indoor volleyball, the objective of the game is to send the ball over the net and to ground it on the opponent's side of the ...
. * ITF: Regulates the height of the tennis ball bounce when dropped on a "smooth, rigid and horizontal block of high mass". Different types of ball are allowed for different types of surfaces. When dropped from a height of , the bounce must be for Type 1 balls, for Type 2 and Type 3 balls, and for High Altitude balls. This roughly corresponds to a COR of 0.735–0.775 (Type 1 ball), 0.728–0.762 (Type 2 & 3 balls), and 0.693–0.728 (High Altitude balls) when dropped on the testing surface. * ITTF: Regulates the playing surface so that the
table tennis ball Table tennis, also known as ping-pong and whiff-whaff, is a sport in which two or four players hit a lightweight ball, also known as the ping-pong ball, back and forth across a table using small solid rackets. It takes place on a hard table div ...
bounces approximately 23 cm when dropped from a height of 30 cm. This roughly corresponds to a COR of about 0.876 against the playing surface. *
NBA The National Basketball Association (NBA) is a professional basketball league in North America. The league is composed of 30 teams (29 in the United States and 1 in Canada) and is one of the major professional sports leagues in the United St ...
: Regulates the gauge pressure of the
basketball Basketball is a team sport in which two teams, most commonly of five players each, opposing one another on a rectangular Basketball court, court, compete with the primary objective of #Shooting, shooting a basketball (ball), basketball (appr ...
to be between 7.5 and 8.5 
psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet * Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviation ...
(51.7 to 58.6 kPa). *
NFL The National Football League (NFL) is a professional American football league that consists of 32 teams, divided equally between the American Football Conference (AFC) and the National Football Conference (NFC). The NFL is one of the major ...
: Regulates the gauge pressure of the
American football American football (referred to simply as football in the United States and Canada), also known as gridiron, is a team sport played by two teams of eleven players on a rectangular field with goalposts at each end. The offense, the team with ...
to be between 12.5 and 13.5 psi (86 to 93 kPa). * R&A/ USGA: Limits the COR of the golf ball directly, which should not exceed 0.83 against a
golf club A golf club is a club used to hit a golf ball in a game of golf. Each club is composed of a shaft with a grip and a club head. Wood (golf), Woods are mainly used for long-distance fairway or tee shots; iron (golf), irons, the most versatile class ...
. The pressure of an American football was at the center of the deflategate controversy. Some sports do not regulate the bouncing properties of balls directly, but instead specify a construction method. In
baseball Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding tea ...
, the introduction of a cork-based ball helped to end the dead-ball era and trigger the
live-ball era The live-ball era, also referred to as the lively ball era, is the period in Major League Baseball beginning in (and continuing to the present day), contrasting with the pre-1920 period known as the "dead-ball era". The name "live-ball era" comes ...
.


See also

* Bouncy ball *
List of ball games This is a list of ball games and ball sports that include a ball as a key element in the activity, usually for scoring points. Ball games Ball sports fall within many sport categories, some sports within multiple categories, including: * Bat-a ...


Notes


References


Further reading

* * * * * *{{cite journal , last1=Erlichson , first1=Herman , year=1983 , title=Maximum projectile range with drag and lift, with particular application to golf , journal= American Journal of Physics , volume=51 , issue=4 , pages=357–362 , bibcode=1983AmJPh..51..357E , doi=10.1119/1.13248 , url=https://www.forbes.com/sites/stevensalzberg/2013/04/29/the-physics-of-golf-whats-the-ideal-loft-to-hit-the-ball-farthest/#5c150bf26926 , access-date=29 April 2013 Balls Sports rules and regulations Classical mechanics Kinematics Dynamical systems Motion (physics)