Bouncing ball

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]

Forces during flight and effect on motion

The motion of a bouncing ball obeys projectile motion.^[2]^[3] Many forces act on a real ball, namely the gravitational force (F_G), the drag force due to air resistance (F_D), the Magnus force due to the ball's spin (F_M), and the buoyant force (F_B). In general, one has to use Newton's second law taking all forces into account to analyze the ball's motion:

{\begin{aligned}\sum \mathbf {F} &=m\mathbf {a} ,\\\mathbf {F} _{\text{G}}+\mathbf {F} _{\text{D}}+\mathbf {F} _{\text{M}}+\mathbf {F} _{\text{B}}&=m\mathbf {a} =m{\frac {d\mathbf {v} }{dt}}=m{\frac {d^{2}\mathbf {r} }{dt^{2}}},\end{aligned}}

where m is the ball's mass. Here, a, v, r represent the ball's acceleration, velocity, and position over time t.

Gravity

Trajectory of a ball bouncing at an angle of 70° after impact without drag , with Stokes drag , and with Newton drag .

The gravitational force is directed downwards and is equal to^[4]

F_{\text{G}}=mg,

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

{\begin{aligned}\mathbf {a} &=-g\mathbf {\hat {j}} ,\\\mathbf {v} &=\mathbf {v} _{\text{0}}+\mathbf {a} t,\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {1}{2}}\mathbf {a} t^{2},\end{aligned}}

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
${\begin{aligned}a_{\text{x}}&=0,\\v_{\text{x}}&=v_{0}\cos \left(\theta \right),\\x&=x_{0}+v_{0}\cos \left(\theta \right)t,\end{aligned}}$	${\begin{aligned}a_{\text{y}}&=-g,\\v_{\text{y}}&=v_{0}\sin \left(\theta \right)-gt,\\y&=y_{0}+v_{0}\sin \left(\theta \right)t-{\frac {1}{2}}gt^{2}.\end{aligned}}$

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]

{\begin{aligned}H&={\frac {v_{0}^{2}}{2g}}\sin ^{2}\left(\theta \right),\\R&={\frac {v_{0}^{2}}{g}}\sin \left(2\theta \right),~{\text{and}}\\T&={\frac {2v_{0}}{g}}\sin \left(\theta \right).\end{aligned}}

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.

Drag

Air flow around the ball can be either laminar or turbulent depending on the Reynolds number (Re), defined as:

{\text{Re}}={\frac {\rho Dv}{\mu }},

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]

F_{\text{D}}=6\pi \mu rv,

where r is the radius of the ball. This force acts in opposition to the ball's direction (in the direction of $\textstyle -{\hat {\mathbf {v} }}$ ). 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]