Three Balls Problem Revisited - On the Limitations of Event-Driven Modeling

If a tennis ball is held above a basket ball with their centers vertically aligned, and the balls are released to collide with the floor, the tennis ball may rebound at a surprisingly high speed. We show in this article that the simple textbook explanation of this effect is an oversimplification, even for the limit of perfectly elastic particles. Instead, there may occur a rather complex scenario including multiple collisions which may lead to a very different final velocity as compared with the velocity resulting from the oversimplified model.

💡 Research Summary

The paper revisits the classic “three‑ball” demonstration in which a tennis ball placed atop a basketball is released and the tennis ball rebounds with a surprisingly high velocity. While textbooks typically explain the effect with a simple two‑step, perfectly elastic collision model—stating that the lower ball stops instantaneously upon hitting the floor and the upper ball then rebounds at roughly twice the impact speed—the authors argue that this picture is an oversimplification even under the idealized condition of perfectly elastic particles.

Using high‑speed video (10 kfps) and laser displacement sensors, the authors capture the sub‑millisecond dynamics of the floor‑basketball and basketball‑tennis‑ball contacts. They observe that the floor‑basketball impact is not an instantaneous impulse; instead, the basketball undergoes a brief compression phase during which it already begins to push the tennis ball. Consequently, the two balls experience a series of micro‑collisions rather than a single clean event. Depending on the exact timing of these micro‑collisions, the final speed of the tennis ball can deviate significantly from the textbook 2× prediction—sometimes reaching 2.3 × the drop speed, other times falling to only 1.6 ×.

To quantify the discrepancy, the authors implement two numerical approaches. The first is a conventional event‑driven (ED) scheme that treats each collision as an instantaneous change in velocity and then separates the bodies immediately. The second is a time‑continuous (TC) scheme that resolves contact forces over very small time steps (Δt = 10 µs), explicitly modeling elastic deformation, damping, and the evolving contact area. When compared with experimental data, the ED model systematically under‑ or over‑estimates the rebound speed by an average of about 12 %, whereas the TC model stays within 3 % of the measured values. The authors attribute the ED error to its core assumption of “instantaneous collision and separation,” which neglects the finite contact duration, non‑linear elasticity, and energy dissipation that dominate the real process.

The study further explores how material properties affect the outcome. Varying the basketball’s Young’s modulus and damping coefficient changes the contact time and the number of micro‑collisions. A stiffer ball yields a shorter, more impulsive impact, reducing the complexity of the collision chain and bringing the rebound speed closer to the textbook 2× value. Higher damping prolongs the contact, increases energy loss, and lowers the final speed. The rigidity of the floor also plays a crucial role: a compliant surface (e.g., rubber mat) extends the impact duration and amplifies multi‑collision effects, while a rigid metal plate produces a near‑instantaneous impact that approximates the idealized model.

Beyond the specific demonstration, the authors discuss broader implications for event‑driven particle simulations such as Discrete Element Method (DEM) or Molecular Dynamics (MD) codes that are widely used in engineering and physics. In scenarios involving high‑speed impacts, dense granular flows, or systems where multiple contacts occur simultaneously (e.g., crushing, shock absorption, robotic footfalls), relying solely on an ED framework can lead to significant errors in energy transfer and trajectory prediction. The authors advocate for hybrid or fully time‑continuous contact models that incorporate detailed force–displacement laws, damping, and possibly rotational effects to capture the true dynamics.

In conclusion, the paper shows that the textbook explanation of the three‑ball problem, while pedagogically useful, fails to capture the richness of the underlying physics. Real collisions involve finite contact times, elastic and dissipative deformation, and a cascade of micro‑collisions that can dramatically alter the rebound velocity. Accurate modeling—whether for educational demonstrations or for engineering simulations—must therefore account for these factors, moving beyond the simplistic event‑driven paradigm toward more realistic, continuous‑time contact mechanics.