Executing a successful third shot drop in pickleball

Abstract

We define and investigate a successful third shot drop in pickleball using a numerical model of pickleball ball aerodynamics. Our overall objective is to investigate the ranges of initial speeds, angles and spins that result in a successful third shot drop. We conclude that the initial speed must be in the range 10.9 metres per second to 13 metres per second for down-the-line shots and 13.3 metres per second to 16 metres per second for cross-court shots. The initial angle must be in the range 15.5 degrees to 22.5 degrees for down-the-line shots and 12.5 degrees to 18 degrees for cross-court shots. We conclude that the effects of spin on the third shot drop are of secondary importance. We believe these results could be useful as a guide to coaches and players wanting to develop this crucial aspect of the game of pickleball.

Introduction

Pickleball is reportedly the fastest growing sport in North America as reported by Fischetti, ¹ and is increasingly being played internationally. The popular press has taken notice, for example: Sison ² and Wollaston, ³ and there is a flood of online videos illustrating the game, as played by both professionals and amateurs. Reasons for the popularity of pickleball include: low levels of athleticism required for entry-level play, relatively low costs of equipment and clothing, high levels of sociability among amateur players, and a ball game that is filled with fast-paced action and high levels of excitement. Brandt ⁴ on the Play Pickleball web site suggests that pickleball could be introduced to the Olympics for the 2032 summer games.

Pickleball is a “racket and ball” game played in either singles or doubles on a court much smaller than a tennis court - 5.2 m (20 feet) ^a wide with baseline to net distance of 6.71 m (22 feet). The net is slightly lower than a tennis net - 0.914 m (36 inches) at the sidelines and 0.863 m (34 inches) at mid-court. Without loss of generality, this paper is structured around a doubles match. The racket, called a paddle in pickleball, has a rigid face and the ball is of low-bounce hard plastic with between 26 and 40 holes for indoor and outdoor play respectively. All measurements and calculations reported in this paper are based on a 40-hole outdoor ball. An important feature of the pickleball court is the Non-Volley Zone, commonly called the kitchen, which is 2.13 m (7 feet) from the net. As the name implies, the ball cannot be volleyed while the player is inside this zone.

The main objective of this study is to present useful advice to both players and coaches regarding the characteristics (initial speed, angle and spin) of a played stroke that leads to a successful third shot drop (after the serve and return-of-serve). To achieve this, we develop a mathematical model of the flight (trajectory) of a pickleball and use the model to investigate the third shot drop.

The game of pickleball

In an interesting deviation from related racket and ball games played on a court, the rules of pickleball are designed to reduce the power of the serve, and to eliminate the possibility of the serve and volley sequence so common in tennis. This is achieved by requiring that the serve be struck underhand from behind the baseline, and must bounce between the kitchen line and baseline. In addition, the return-of-serve must be struck only after a bounce. As a result, the third shot in a rally is often struck near the baseline. The third shot cannot be volleyed, a condition that does not hold for future shots. A common strategy for the third shot is to play it as a drop shot that lands inside the opposing kitchen. This is called a third shot drop. It is a shot often practised in drills by beginner players, and almost universally used in competitive play at all levels. The third shot drop is a much-discussed aspect of pickleball play, and is the subject of a multitude of freely-available training videos. A prominent example is given by Koszuta. ⁵ As in the majority of similar videos, the author makes recommendations about player positioning after the serve, paddle grip and swing, and very general advice about the desired ball trajectory. A universal piece of advice is to achieve a trajectory that reaches an apex before crossing the net, and to aim for a bounce in the far part of the opposing kitchen. The conventional wisdom among pickleball players is that doing so reduces the options available to opposing players when playing the fourth shot of the rally. These features of the third shot drop are well explained by Nard et al., ⁶ who also outlines general stroke techniques for making the shot. Balls that bounce in the far part of the opposing kitchen generally cannot be attacked, but rather force the receiving player to play a slow ball that barely crosses the net (called a dink shot). This defensive play is a common feature of all pickleball play among intermediate and stronger players.

Because the game of pickleball is so recent, very little work has been done on technical aspects of the play. One recent book Swartz ⁷ addresses various aspects of pickleball through statistical analytics. However, not much is known about the aerodynamic characteristics of the pickleball ball. This will require us to start our work by an analysis of pickleball ball flight so as to determine the two main aerodynamic characteristics of ballistic objects: the drag- and lift-coefficients. ^b

Modelling pickleball trajectories

There exists a wide and varied scientific literature on the aerodynamics of sports balls. An early general review of the aerodynamics of balls in a wide range of sports was provided by Mehta. ⁸

Specific sport balls have been studied by: Asai et al. ⁹ for soccer balls, Alaways and Hubbard ¹⁰ for baseball, Bearman and Harvey ¹¹ and Mizota ¹² for golf, and Cross and Lindsey ¹³ and Ivanov ¹⁴ for tennis. While many studies have examined 2-D ball trajectories, Ivanov ¹⁴ and Mizota ¹² are notable in that they have examined 3-D trajectories. Most studies of ball flight in sports use aerodynamic engineering methods, combined with force-balance numerical models. Wen et al. ¹⁵ provides a notable recent advance in the use of deep-learning or Artificial Neural Network methods to analyze baseball trajectories.

Explicit modeling of the aerodynamics of a pickleball ball has been carried out by Creer, ¹⁶ who used computational fluid dynamics modelling in an attempt to estimate lift- and drag-coefficients of 26- and 40-hole pickleballs. The work was inconclusive because of computational difficulties. More relevant to our present work is the study of Emond et al. ¹⁷ who model pickleball flight trajectories in order to investigate the advantage or disadvantage of upwind versus downwind play. This work does not consider aerodynamic lift, and uses a drag coefficient of $C_d=0.6$ based on measured drag coefficients of a Wiffleball. The work emphasizes the need for measurements to provide a direct estimate of $C_d$ for pickleball balls.

The majority of these studies present conclusions that are only useful to players and coaches with unusually deep technical backgrounds. Our objective is to conduct modelling of the flight of a pickleball and provide an interpretation of the results that will be useful for players and coaches of the game. Our interests thus go beyond ball aerodynamics to focus on a strategy for executing a successful third shot drop.

Our development of the pickleball ball flight model, and its use to determine the two characteristics that quantify aerodynamic lift and drag (lift- and drag-coefficients, ( $C_d$ and $C_l$ respectively)) for a pickleball ball are shown in the Appendix. Our model is based on well-known equations for the aerodynamics of flying, spinning objects. Measured trajectories of a pickleball ball in flight are used to determine the lift- and drag-coefficients. This model is then recast in an exploratory mode to investigate the dependence of the ball trajectory on the three strike parameters: initial speed, strike angle and spin rate. The initial conditions in our modelling will represent a typical game context in which the third shot is struck from near the baseline $x=0.0\mathrm{m}$ and at a height of roughly $z=0.75\mathrm{m}$ . In a real game of pickleball, these two initial conditions are likely to vary. Modelling a reasonable range of strike height and position is entirely possible, but will not result in greatly different overall conclusions. Our requirement for the third shot drop is that the ball clears the net, and bounces somewhere within the kitchen. In the exploratory model, the three strike parameters are specified as run conditions. As described in the Appendix, the spin rate is used to determine the lift coefficient through the spin parameter.

In Figure 1 we show three shots from the model in exploratory mode, with different initial spin rates. These shots illustrate the overall model performance, and show the effect of varying spin. All three shots had the same spin rate of 10 revolutions per second ( ${\mathrm{s}}^{-1}$ ) but the sign (positive or negative) of the resulting lift coefficient was set to reflect the spin sense (backspin or topspin). As is evident from Figure 1 the model captures a strong spin effect on ball trajectory.

Figure 1.

Three modelled pickleball trajectories with backspin ( $C_l=-0.12$ , green trajectory), no spin ( $C_l=0.0$ , blue trajectory) and topspin ( $C_l=+0.12$ , red trajectory). The blue vertical bar depicts the net, “K” indicate kitchen lines, and “B” indicate baselines. In these trajectories, $C_d=0.30$ , initial speed $=12{\mathrm{ms}}^{-1}$ , initial angle $={20}^{\circ }$ , initial spin rate $=10.0{\mathrm{s}}^{-1}$ , spin parameter = 0.19. Reynolds number ranged from $5.3\times {10}^4$ to $6.4\times {10}^4$ .

The following section will explore the idea of a successful third shot drop, and then use the model to understand how such a shot may be achieved through combinations of initial speed, strike angle and spin rate.

The third shot drop

Repeated running of the trajectory model with specified shot speed, angle and spin rate can be used to investigate the third shot drop. It will always be possible to achieve a given total range with two possible shots, one having a high (generally greater than 45 degrees) shot angle and the other a much lower shot angle. The high angle shot is of no interest since such shots will give opposing players a lot of time to react. Furthermore, a high angle shot will result in a high bounce, giving the opposing players an opportunity to attack the ball. We therefore only consider the low-angle shots. There exist two possible limiting shots: one that just clears the net and bounces somewhere in the kitchen, and the other that clears the net widely and bounces exactly on the kitchen line. Figure 2 shows the result of this exploration for shots made at 0.75 m high, on the baseline. A number of features are worth noting:

At speeds below about $8.7{\mathrm{ms}}^{-1}$ it is not possible to clear the net with shots from the baseline.

At speeds higher than about $13.5{\mathrm{ms}}^{-1}$ the ball will always land beyond the kitchen line.

Topspin shots with a spin rate of $10{\mathrm{s}}^{-1}$ and strike angles greater than 35 degrees will not clear the net. For backspin strikes, the limiting angle is 33 degrees.

The convergent lines at $13.5{\mathrm{ms}}^{-1}$ indicate trajectories of balls that bounce on the kitchen line.

Shots that fall in the space between “top10K” and “back10” can be considered legitimate third shot drops only in the sense that they land between net and kitchen line. In the coming analysis we will narrow our consideration of all third shot drops to consider only those that can be considered successful.

We plot lines only for spin rates of $10{\mathrm{s}}^{-1}$ topspin and backspin and zero spin. Spin rates lower than $10{\mathrm{s}}^{-1}$ fall at intermediate positions and make for a very crowded figure. Spin rates up to $15{\mathrm{s}}^{-1}$ are noted as possible by Lindsey, ¹⁸ limited only by ball and paddle roughness. We judge such extreme spin rates to be unlikely in most play by recreational pickleball players.

Figure 2.

Combinations of angle, speed and spin rate that result in third shot drops that land in the kitchen. “top” (red lines) and “back” (green lines) indicate topspin and backspin shots respectively. “K” indicates shots that land on the kitchen line. We plot only spin rates of $10{\mathrm{s}}^{-1}$ , as lower spin rate shots will fall between the respective lines. Blue lines are for zero spin rate.

Figure 2 depicts all possible low-angle shots that clear the net and bounce in the kitchen with down-the-line (parallel to court sidelines) trajectories. As discussed, the strategically strongest third shot drops are ones that bounce somewhere close to the kitchen line in order to prevent the serve receivers from making the fourth shot an attack shot. For convenience we model third shot drops that bounce beyond three quarters way between net and kitchen line, but short of the kitchen line. We also model only shots that barely clear the net. This is to ensure that the ball’s (downward) vertical velocity is a minimum possible, and so the bounce height of the ball is also a minimum. These conditions are consistent with the general guidance given by Koszuta ⁵ and Nard et al. ⁶ It is also important to consider cross-court third shot drops. For these reasons we develop a figure that is similar to, but a subset of Figure 2, and include both down-the-line and cross-court shots. For completeness, we include topspin shots with spin rates of $15{\mathrm{s}}^{-1}$ , and assume that backspin rates this high will be difficult to achieve. These are shown in Figure 3.

Figure 3.

Combinations of angle, speed and spin rate that result in third shot drops that would be strategically strong, landing near the back one quarter of the kitchen. Red/green lines define ranges of speed and angle for down-the-line/cross-court shots with spin rates of $10{\mathrm{s}}^{-1}$ . Lower spin rate shots will fall between the respective topspin and backspin lines. Black lines show shots with topspin of $15{\mathrm{s}}^{-1}$ , blue lines show shots with no spin.

An examination of Figure 3 reveals the following features that are relevant for players making and developing their skills at executing a third shot drop, and for coaches instructing players on the shot:

The range of ball speeds for a successful third shot drop is $10.9{\mathrm{ms}}^{-1}$ to $13{\mathrm{ms}}^{-1}$ for down-the-line shots and $13.3{\mathrm{ms}}^{-1}$ to $16{\mathrm{ms}}^{-1}$ for cross-court shots. These are relatively narrow speed ranges, and while it is unlikely even expert players will be able to calibrate their shot speed with this precision, it does suggest that developing skill in shot speed control is very important in making a successful third shot drop.

The range of shot angles for a successful third shot drop is 15.5 degrees to 22.5 degrees for down-the-line shots and 12.5 degrees to 18 degrees for cross-court shots. As with shot speed, these are relatively narrow shot vertical angles, and suggest developing skill in shot angle control is very important in making a successful third shot drop.

The effects of varying speed and angle are much stronger than the effect of varying spin. This is evident in Figure 3 from the relatively small differences in speed and angle that result from extreme variation between topspin and backspin. Spin variation between a topspin of $15{\mathrm{s}}^{-1}$ and backspin of $10{\mathrm{s}}^{-1}$ produce a trajectory variation less than that produced by a speed variation of about $0.8{\mathrm{ms}}^{-1}$ for down-the-line shots and roughly $1.0{\mathrm{ms}}^{-1}$ for cross-court shots. These observations suggest that players wanting to execute successful third shot drops should concentrate on angle and speed control, rather than on spin development.

Conclusions

Expert pickleball players and coaches agree that the trajectory of a successful third shot drop in pickleball reaches its apex before crossing the net, and bounces in the far portion of the opposing kitchen. These characteristics give the serving side the maximum chance of continuing in a successful rally.

We investigated the third shot drop in pickleball, using a numerical model of pickleball ball flight. The lift- and drag-coefficients of a 40-hole pickleball ball were determined by fitting modelled to measured pickleball ball trajectories, as detailed in the Appendix. Using the model in an exploratory mode, we were able to determine ranges of initial speed, angle and spin that resulted in trajectories producing a successful third shot drop.

We conclude that:

The initial speed must be in the range $10.9{\mathrm{ms}}^{-1}$ to $13{\mathrm{ms}}^{-1}$ for down-the-line shots and $13.3{\mathrm{ms}}^{-1}$ to $16{\mathrm{ms}}^{-1}$ for cross-court shots.

The initial angle must be in the range 15.5 degrees to 22.5 degrees for down-the-line shots and 12.5 degrees to 18 degrees for cross-court shots.

The effects of spin on the third shot drop are of secondary importance (relative to the influence of speed and angle) in determining where the ball bounces. We acknowledge that ball spin can have a strong effect on the post-bounce trajectory.

We acknowledge that speed and angle control at the level of precision noted above can only be achieved with considerable practice and skill. Using immediate playback video recording of shots in practice sessions to determine initial speed and angle could be a beneficial coaching tool to achieve such shot control.