Physics of Bowling: Simulation and Strategy

The problem

A bowling ball is a rigid body whose centre of mass and principal axes are deliberately offset from its geometric centre. Thrown with spin onto a lane that is oiled unevenly, it first skids, then hooks as friction builds, then rolls. Where it ends up, and at what angle it enters the pocket between the head pin and the three pin, decides whether the shot is a strike. The whole problem is Euler’s equations plus a friction law that depends on where the ball is on the lane, which makes it a clean, complete piece of classical mechanics that is not solved in any textbook and is genuinely useful to people who bowl.

Basis paper

S. S. M. Ji, S. Yang, W. Dominguez, C. G. Hooper and C. S. Bester, “Using physics simulations to find targeting strategies in competitive tenpin bowling”, AIP Advances 15, 045222 (2025), doi:10.1063/5.0247761. Open access under CC BY.

The authors derive six coupled ordinary differential equations for the ball from Euler’s equations for a rotating rigid body, integrate them numerically on a lane with a realistic oil pattern, show the three phases of motion, and produce a map of the initial conditions (launch position, angle, speed, rev rate) that lead to a strike. They then model an imperfect bowler by adding variance to the launch and show that some targets give a higher strike rate than others because the oil pattern creates “miss room”.

The path

1. Study. Rigid-body dynamics beyond the degree course: Euler’s equations in the body frame, asymmetric inertia tensors, the kinematics of a rolling and slipping sphere, and dry friction with a coefficient that varies in space. Numerical integration of stiff, event-driven systems (the skid-to-roll transition) with scipy.integrate.solve_ivp. Monte Carlo sampling of launch variance.

2. Reproduce. Implement the equations of motion in Python and reproduce, one by one, the paper’s results: a single trajectory showing skid, hook and roll; the entry position and angle at the pins for a grid of launch conditions; the strike map; and the strike-rate map for the imperfect bowler. Agreement with the published figures is the checkpoint before anything new is attempted.

3. Find the gaps and go beyond. The paper fixes one oil pattern, one ball and one criterion for a strike. From there:

• run the model on the standard competition oil patterns published by the sport’s governing bodies and compare the strike maps;
• vary the ball’s mass distribution (radius of gyration, differential) and see how the hook phase changes;
• replace the pocket-entry criterion with a simple pin-collision model and compare;
• study how sensitive the optimum target is to the bowler’s variance, and whether the “miss room” result is robust;
• if time allows, fit the friction law to a published lane-friction measurement instead of assuming it.

What you need

Classical mechanics and differential equations from the degree; Python with NumPy, SciPy and Matplotlib. No laboratory work; the “experiment” is the simulation and its comparison with the published data. Suitable for one student.