(
FG), the
drag force (
FD), the
Magnus force (
FM), and the
buoyant force (
FB). The motion of a bouncing ball obeys
projectile motion. 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: :\begin{align} \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{align} where
m is the ball's mass. Here,
a,
v,
r represent the ball's
acceleration,
velocity, and
position over
time t.
Gravity The gravitational force is directed downwards and is equal to :F_\text{G} = mg, where
m is the mass of the ball, and
g is the
gravitational acceleration, which on
Earth 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{align} \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{align} 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 (
R) and
time of flight (
T) of a ball bouncing on a flat surface are given by :\begin{align} 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{align}
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 D v}{\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 , and . If the Reynolds number is very low (Re F_\text{D} = 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{v}). 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{D} = \frac{1}{2} \rho C_\text{d} A v^2, 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.
Magnus effect . The curly flow lines represent a
turbulent wake. The airflow has been deflected in the direction of spin. 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 :F_\text{M} = \frac{8}{3} \pi r^3 \rho \omega v, 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 \textstyle \hat \mathbf{\omega} \times \hat \mathbf{v}). 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{M}=\frac{1}{2}\rho C_\text{L} A v^2, 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. In certain conditions, the lift coefficient can even be negative, changing the direction of the Magnus force (
reverse Magnus effect). 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. In
baseball,
pitchers use the effect to create
curveballs and other special
pitches.
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. 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.
Buoyancy Any object immersed in a
fluid such as water or air will experience an upwards
buoyancy. 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{B} = \frac{4}{3}\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==