Using The Beach Ball Physics Model

The physics of a thrown beach ball can be very interesting. Because of
its relatively low mass (compared to its size), subtle effects that are
important in sports like baseball, soccer and volleyball become
exaggerated in the beach ball’s motion. This simulation allows users to
explore a simple model of several “complications” to simple projectile
motion: drag, buoyancy and the Magnus effect. Drag slows the ball down,
buoyancy makes the ball “float” and the Magnus effect puts a curve on
the ball’s trajectory.
Simulation Controls:
The simulations main display includes an
interactive 3-D display of the ball in flight. The “Show Velocity”
checkbox will add a pair of vectors for the ball’s velocity and angular
velocity (both scaled to the ball’s launch speed). The “Show Forces”
checkbox adds vectors for all the forces contributing to the ball’s
dynamics. A pair of graphs may be hidden or viewed for ancillary
information using the Show Graphs checkbox.

The basic ball launch parameters are the initial height of the ball, the
launch speed and the launch angle α (from horizontal).
The rotation of the ball is determined by the spin (in rotations per
second) and the orientation of the spin axis. The axis starts out with a
horizontal orientation lined up with the plane of the original “forward”
motion of the ball. The axis is reoriented by specifying the tilt (up or
down) relative to horizontal and an azimuthal angle corresponding to
rotating the spin axis about a vertical axis. For no spin effects simply
set the spin to 0. For a simple curve ball, set the spin (1 to 5 rotations
per second are fairly typical) and set the spin axis to a vertical
orientation with a tilt of ±90 degrees and an azimuth of 0. For a
floating effect due to backspin, set the spin rps, set the tilt to 0 and
set the azimuth to -90 degrees, while a top spin sinker can be
demonstrated setting the azimuth to + 90 degrees.
Beach Balls come
in various sizes, so the radius and mass of the ball can be specified.
Note that the assumption is that the mass given is what would be measured
on a balance; the mass of the air inside the ball does not affect such
measurements as its weight is offset by the buoyancy due to the displaced
volume of air. To see the buoyant force in a simple way, note that if you
weigh your beach ball empty or inflated, you will get the same measurement
and yet the inflated ball’s contents do contribute to the overall inertia.
The
simulation also allows the adjustment of some of the aerodynamic
parameters. See the technical notes for discussion. For those interested
in the aerodynamics, the Reynolds number and Spin factor are also
displayed.
The Physics of Beach Ball Trajectories: Technical Notes
The model employed in this simulation follows Clanet’s “Sports Ballistics”.1
This simulation takes up as the positive z direction and the initial
velocity of the ball to be in the y-z plane.
Gravity and Buoyancy
In
a Physics Teacher paper on the vertical motion accounting for aerodynamic
effects, Timmerman and van der Weele2 note
“… we include the buoyant force, which is almost always forgotten or
ignored in the literature on this topic. This is somewhat curious, since
the medium that is responsible for the drag force at the same time
introduces a buoyant force.”
The mass (and hence the weight) of a beach ball are easy to determine:
just place it on an electronic balance. The same results will be obtained
whether the ball is inflated or not. When a ball is thrown, the air
inside goes for a ride as well, and in the case of a beach ball, this is a
significant contribution to inertia:
The weight is proportional to this total mass:
The weight is partially offset by the buoyant force, which is the weight
of the fluid displaced. The displaced fluid is air, at the about the same
density as the air inside the ball (which is only lightly compressed).
Thus the buoyant force is
For most balls used in sports (baseball, soccer, volleyball, etc.) the
buoyant force and the inertia of the air contained within the ball are
ignorable.
Drag Force
Drag force, or air resistance is
the friction effect of moving though the air. It will be a significant
force for almost any toss of the beach ball. The drag force is opposes
motion and it is modeled in this simulation by

,
where
Clanet cited typical values for the drag coefficient to be about .25 for
the balls used in soccer, volleyball and basketball. Some local
experiments at Penn State Schuylkill using video analysis of the terminal
velocity of beach balls suggest a typical closer to .5 for a beach ball’s
drag coefficient, and the idealized smooth sphere would have a drag
coefficient of .48.3
Magnus Effect
The
Magnus force is very noticeable when it is used to make the trajectory of
a ball curve while in flight. The size of the effect depends upon both
the velocity and angular velocity of the ball, and the direction is always
perpendicular to both the velocity and the rotation axis of the ball’s
spin. The force is often referred to as lift, and (following Clanet) is
given by:
where C_spin is a dimensionless parameter which for Clanet is about 1.7
when the spin factor
An alternate (and more common) formulation of the Magnus force is given
by
which, with the cross section area A=π r2 and the lift
coefficient CL=2 CD S, the Magnus force would
be then be written in vector form as:
Which corresponds to Clanet form and typical Cspin when CD≈.5.
Complete
Model Dynamics
The full beach ball dynamics used in the model is
given by
An important caveat is that the drag and lift models tend to get more
complicated at the well-known “drag crises” near Reynolds number Re=105.
Bottom line: as sophisticated as this model is compared to drag-free
ballistic motion, one should still not necessarily expect high fidelity
results over all launch conditions.
(1)
Sports Ballistics by Clanet, Christophe, Annual Review of Fluid Mechanics,
01/2015, Volume 47, Issue 1
(2) On the rise and fall of a ball with
linear or quadratic drag by Timmerman, Peter; Weele, van der, Jacobus P,
American Journal of Physics, 1999, Volume 67, Issue 6
(3) For more
information on drag, see https://en.wikipedia.org/wiki/Drag_coefficient
(4)
Nathan AM. 2008. The effect of spin on the flight of a baseball. Am. J.
Phys. 76:119–24