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Using The Beach Ball Physics Model

A Beach Ball in Flight 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.

Launch Parameters The basic ball launch parameters are the initial height of the ball, the launch speed and the launch angle α (from horizontal).

Setting the Beach Ball Rotation
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
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
(4)  Nathan AM. 2008. The effect of spin on the flight of a baseball. Am. J. Phys. 76:119–24  


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Software Requirements


Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support full-screen? Yes. Chrome/Opera No. Firefox/ Samsung Internet Not yet Yes Yes
cannot work on some mobile browser that don't understand JavaScript such as.....
cannot work on Internet Explorer 9 and below



Michael R. Gallis, Ryan Vidal; lookang

Sample Learning Goals


For Teachers





 Backspin Basketball Flies Off Dam by Veritasium


  Physics of toys- Cup Flyers // Homemade Science with Bruce Yeany by Bruce Yeany



Question 1: A student claims that having a backspin on the ball increases the distance travelled by the ball as compared to having a top spin. True or false?

True. Backspin causes the ball to travel further than with having a top spin.

Question 2: Taking the default point of reference from the sim (blue arrow points up, red arrow points towards user, and green arrow towards right of user), how does having a left spin affect the trajectory of the ball?

Ball moves to the right and also away from the user.

Question 3: If a tennis player hit the ball with a certain spin and it reached the ground faster than expected, what type of spin did the ball experience? (assuming the tennis ball in this case is modelled by the same ball in the sim)

Top spin. This exerts a downward force on the ball and thus makes it hit the ground faster.

Other Resources


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