Using physics simulations to find targeting strategies in competitive tenpin bowling

Tenpin bowling remains as one of the most popular sports in the USA, with over 45 × 10⁶ people participating regularly as of 2017.¹ With millions of dollars at stake every year in national competitions, significant research has been done to understand how players can achieve higher scores. Due to the complexity of the calculations and the vast number of variables that can affect the ball’s trajectory, most of the research has focused on statistical analysis from empirical data instead of theoretical modeling. For example, the 2018 US Bowling Congress (USBC) Equipment Specifications Report used “37 bowlers with a full range of revolutions per minute (RPM) rates” rather than a computer model² and performed an analysis of ball tracking data from a professional bowling tournament.³ 

The literature on quantitative analysis of bowling physics is rare due to the many parameters involved but has been attempted by Fröhlich, Hopkins, and Huston over the past few decades.^4–6 Fröhlich and Huston created mathematical models that took into account the effects of a weight block within the bowling ball and provided simulation results for a small sample of parameter values, including effects of varying radius of gyration (RG), center of gravity (CG) offset from the geometric center of ball, and initial angular velocity. The simulations demonstrated the qualitative effects of changing certain variables but only assumed simple friction profiles that arise from the distribution of oil that is applied to the lane, known as an oil pattern.

Bowlers will typically look to play in an area of the lane such that they have a small region around their intended target that will result in their ball still hitting the headpin in the desired area at the desired angle so that the strike percentage remains high. This will allow for slight inaccuracies in their shot to not be punished greatly in terms of score. It has been shown that the optimal location for the ball to hit the headpin is $∼6$ cm offset from the center, and the optimal entry angle for the ball to be incident to the pin at is $∼6$°.⁷ 

This paper aims to demonstrate target strategies that can be implemented by a bowler through a simulation that samples a large number of possible initial conditions, and it explores the effects of realistic friction profiles based on oil patterns that are used in current leagues and tournaments.^8,9 The program can then provide the user with the best possible starting position, given their personal bowling numbers, such as ball speed, axis rotation, axis tilt, and angular velocity, by using experimental data to predict the outcome of the trajectories given by the mathematical model.

The equations of motion derived in this section describe rigid body rotation using a rotating frame of reference fixed to the ball. The derivation uses an approach through Euler’s equations, where they are only dependent on the principal moments of inertia and not on the non-diagonal terms in the inertia tensor. This is important as the full inertia tensor of a reference frame fixed to space cannot be easily determined simply based on the published radius of gyration (RG) and differential values of each ball. Previous theories derived by Hopkins assumed a perfectly uniform spherical ball,⁵ while theories from Fröhlich required knowing the non-diagonal terms in the inertia tensor of the bowling ball.⁴ The CG offset, which is typically $∼1$ mm away from the geometrical center of the ball, is assumed to be zero for this study in order to simplify calculations.

In competitive bowling, oil is applied to the lanes in patterns specifically designed to create challenging friction profiles that leave varying margins for error.^8,9 Sample oil patterns are illustrated in Fig. 3. During competition, bowlers have to make educated guesses with shot features, such as ball choice, ball speed, and where to play on the lane, in order to maximize their chance of striking.

Two of the most significant factors that contribute to the chance of a strike are the entry position and entry angle. Entry position is the $x̂$ coordinate of the ball’s position at the moment it makes impact with the pins (⁠$ŷ≈60$ ft). Entry angle is the angle that the ball trajectory makes relative to the vertical at $ŷ≈60$ ft. The ideal position of entry is between 4 and 12 cm (1.57 and 4.72 in. or 1.5 and 4.4 boards) offset from the center of the lane, typically referred to by bowlers as the “pocket.” A 2009 empirical study conducted by the USBC provided the strike percentages at various entry angles and positions.⁷ These data are incorporated into the simulation, allowing it to calculate a score based on the ball’s trajectory.

In real life, no bowler can hit their target with 100% accuracy. The best professionals can get to within around 0.1° from their intended initial launch angle, which can correspond to a difference of a few centimeters further down lane. As shown in Fig. 4, this change will significantly affect the chances of a strike. In this study, the inaccuracy is modeled using a Gaussian distribution of starting angles, with the standard deviation equal to 0.1°.

The equations of motion are solved subject to the following initial conditions that are typical for a competitive bowler. It is assumed that the bowler releases the ball at the foul line (y₀ = 0 m); the angular velocity that they impart to the ball at release is 416 rpm, that is rotating about an axis aligned 45° to the $x̂$ axis and 13.3° to the $x̂−ŷ$ plane, which gives ω_x,0 = −30 rad/s, ω_y,0 = −30 rad/s, and ω_z,0 = 10 rad/s; and the initial speed of the ball is v₀ = 8 m/s (17.9 mph or 28.8 kph). The parameters x₀ and θ₀ are varied to find the different strike results. The properties of the bowling ball itself are as follows: RG = 6.35 cm, Diff. = 0.1 cm, Int. Diff. = 0 cm, and m = 6.8 kg.⁴ 

The motion of the bowling ball on the lane can be categorized into the two classical phases of sliding and rolling. During the sliding phase, which typically occurs when the level of friction is low, the contact point between the ball and lane is not stationary in the lane’s frame of reference (v_b ≠ 0). Due to the low friction and large rotational inertia imparted by the bowler during the motion of releasing the ball, the ball spends most of its travel path in the sliding phase. Once the surface speed decreases to zero, pure rolling phase begins as the contact point becomes stationary relative to the lane. The ball travels in a straight line toward the pins as no more torque is applied. Figure 5 demonstrates this through a plot of v^b over time on a flat oil pattern, where the distribution of oil is uniform throughout the area of the lane that is oiled.

In order to find the initial position x₀ and launch angle θ₀ that can lead to a strike, the outcomes for all possible x₀ values (0–39 boards) and starting angles θ₀ between 0° and 6° relative to the y axis were calculated. The results of this simulation are shown in Fig. 6 for the 40 ft flat oil pattern.

The simulation results can then be combined with the strike chance plots in Fig. 4, to determine the exact chances for these “pocket hits” in the black region to result in a strike. In order to incorporate the “imperfect bowler” idea from Sec. II C, a weighted average is calculated based on a Gaussian distribution of starting angles.

Figure 7 shows that there does not seem to be a strong preference toward any starting position for the flat pattern, with only a narrow region of possible starting conditions all exhibiting similar shades, which correspond to a similar probability of getting a strike. For the short pattern, however, a much wider area of viable starting positions can be found to have a high strike percentage near the turning point of the v-shaped probability curve, which arises from the inhomogeneous oil distribution near the edge of the lane. The highest strike chance is also found in this region, with an estimated value of 0.89, much larger than 0.75 found in the flat pattern. We define the “ideal line” on the short pattern to be the starting board and angle found by the simulation: x₀ = 28.0 boards; θ₀ = 1.8°.

The significance of the “maximum strike chance” is more clearly shown in Fig. 8. Even though both the “ideal line” (x₀ = 28 boards, θ₀ = 1.8°) and the “non-ideal line” (x₀ = 10 boards, θ₀ = 3.2°) lead to a pocket hit, small deviations in starting angle due to human error results in a much larger deviations in the final entry position for the “non-ideal line” (8 cm, compared to less than 3 cm for the ideal line).

The ideal line forms a path that closely aligns with the boundary between two friction zones of the short oil pattern. This makes sense qualitatively because if a bowler misses slightly to the right, the higher friction near the gutter would accelerate the ball to the left. Similarly, the lower friction in the center means a shot that misses slightly to the left will not hook early. In playing in this area, the bowler has some region around their intended target that they can hit while still getting the ball to finish in the desired region at the pins. While this result is not surprising to experienced bowlers, new bowlers often struggle to understand this concept and make adjustments that leave them with a much smaller margin for error.

We have derived equations of motion that describe the evolution of a bowling ball with a fixed rigid internal weight block that travels on a lane with variable levels of friction that arise from the initial distribution of oil known as the oil pattern. For a given set of initial conditions that are typical of a competitive bowler, the equations of motion are solved for two different oil patterns, and an optimal strategy is formed that yields the largest target size while maintaining a high probability of getting a strike by matching the features of the ball path to experimental data. These results are easier to obtain than empirical studies and will be useful for both competitors and their coaches, as well as tournament oil pattern designers who would be able to gauge what the scoring pace of the tournament would be, given a group of bowlers.

In the theory derivations, several important factors that impact the ball’s motion could be considered for future study. For example, the model could be extended to capture the topography of the lane that could also influence optimal strategy. Such features are neglected here as it is assumed that the lane is perfectly flat. The hardness of the bowling ball could also be captured by not assuming that the bowling ball contacts the lane at a point.

Further numerical investigation could also be performed to gain insight into the effect of features such as ball speed and axis rotation on optimizing the margin for error. Further experimental work is required to provide a more accurate set of friction coefficients that are specific to bowling ball surface, type of oil, and lane surface.

The model can be used to help bowlers choose the balls they would bring to the competition, as well as allowing oil pattern designers to determine whether they created a profile that is challenging but also fun to play in.

The authors have no conflicts to disclose.

S. S. M. Ji: Conceptualization (lead); Methodology (lead); Software (lead); Writing – original draft (lead). S. Yang: Conceptualization (equal); Formal analysis (equal); Project administration (lead); Writing – original draft (equal); Writing – review & editing (lead). W. Dominguez: Conceptualization (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (supporting). C. G. Hooper: Conceptualization (equal); Formal analysis (equal); Project administration (lead); Writing – original draft (equal); Writing – review & editing (lead). C. S. Bester: Conceptualization (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (supporting).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.