The Physics and Timing of the Outfield Bounce Throw

Not every great throw from the outfield has to be on the fly. (via Adam Baker)

Editor’s Note: This piece was adapted from a presentation of the same name at SaberSeminar 2018. The presentation slides can be found here.

Last year, I explored a simple question inspired by episode 1041 of Effectively Wild. While hosts Ben Lindbergh and Jeff Sullivan, with some input from Dr. Alan Nathan, quickly decided it was unlikely a ball could gain speed if it bounced off the infield turf, I wondered if there were situations in which a bounced throw could be quicker for an infielder throwing across the diamond, rather than trying to reach the first baseman on the fly.

I began by reviewing the literature of the relevant physics, and constructed trajectory simulations to answer the question. At SaberSeminar 2017, I presented my conclusion that for longer distance throws, a “bounce throw” could reach first base more quickly than an “air throw,” assuming the same release speed for both throw types. The follow-up article contained more detail on my assumptions and methods, and hinted at my topic for this year’s SaberSeminar: the outfield bounce throw.

Attempted outfield assists, whether to home plate or another base, frequently bounce before reaching their targets, though the most memorable ones often do not, including Ramon Laureano’s on August 11. I don’t have much high level playing or coaching experience, so I don’t know whether there is conventional wisdom around bouncing outfield throws, but I think this is irrelevant to my current inquiry. Instead, I wish to use basic physics and math to make reasonable estimates of the flight times for both types of throw for a range of conditions, and to quantify the differences between them.

The essential idea is that, even though the ball loses kinetic energy and speed to friction — both external from contact with the ground and internal from compression during the bounce — a bounce will allow a lower release angle and shorter total path from thrower to receiver than if the throw were to take a higher arcing path to reach on the fly. In this way, studying outfield throws is no different than infield throws, but allows me to expand the range of throw distances under examination, as well as to adjust some of my methods.

Baseball Aerodynamics

Although covered in greater detail in my original article, a brief review of the relevant physics is again merited. We must begin with the well-established aerodynamics of a baseball in flight, as reported in Dr. Nathan’s article on the home run surge and accompanying technical write up, as well as Rod Cross’ text, Physics of Baseball and Softball.

Figure 1. Forces on a spinning ball in flight including drag (FD), Magnus force (FM), and gravity (FG)

I will assume for the purposes of this analysis that the ball is travelling purely in the vertical plane with speed v, and has pure backspin with angular velocity ω, about an axis normal to that plane. Gravity always acts to accelerate the ball toward the ground, while the drag force acts to slow the ball in its direction of travel. The Magnus, or lift, force acts perpendicular to the direction of travel and is the source of the extra carry of a home run ball hit with backspin, or the downward action of a 12-6 curveball.

Figure 2. Aerodynamic forces on a spinning baseball.

The drag force and Magnus force are both proportional to the density of air, the cross-sectional area of the ball, and the square of its speed, as well as a drag coefficient and lift coefficient respectively. While much has been made about evidence of a systematic change to the average drag coefficient of major league baseballs in MLB’s commissioned home run report, earlier research from Kensrud suggests that a fixed drag coefficient of 0.35 is reasonable to represent the common range of speeds and spins encountered in a professional baseball game. Furthermore, because the present analysis seeks to compare two throws from the same location at the same initial speed, the exact choice of drag coefficient is not critical.

The lift coefficient is known to be a function of the “spin ratio,” which is defined as the product of the radius of the ball and the ratio of its angular and linear velocity. Here, I have assumed the functional form suggested by Nathan and Sawicki in their work on the topic. As for the spin rate itself, I assume that the spin is determined by the release speed of the throw. This relationship is modeled with a linear regression on 2018 pitcher fastball data from Baseball Savant, as shown in Figure 3, where a 5 mph change in release speed results in about 100 rpm change in spin rate.

The Ghost of Passed Balls Past
Even the best catchers miss some.

Figure 3. Relationship between throw speed and backspin.

Calculating the trajectory of a ball in flight requires solving the equations of motion from Newton’s second law with the three forces in Figure 2 in both the vertical and horizontal directions. While not particularly complex, these types of nonlinear differential equations are commonly solved by numerical integration. I take the same approach here, using the fourth-order Runge-Kutta method (RK4). All this calculation is done in the Python language using the essential Numpy and Scipy libraries.

Bounce Physics

For baseballs bouncing at relatively oblique angles, the relevant bounce model is one in which the surface of the ball slides throughout the bounce, as opposed to gripping it at any point. Again, I use the model presented in Cross’s text.

Figure 4. Diagram and equations of the bounce model.

The final linear velocity — both the horizontal and vertical components — as well as the final angular velocity, are related to the incident values by the vertical coefficient of restitution (COR, ey) and the dynamic coefficient of friction, μ, of the ball-surface interface. Together, these quantities determine how much of the ball’s speed is lost in the collision and how much the spin changes. While literature exists that attempts to quantify these parameters for particular surfaces, I will simply adopt the values introduced by Cross, where COR = 0.55, and μ = 0.4. Programmatically, while the trajectories are being calculated, I determine the time step at which the ball “hits” the ground, and apply the equations above to populate the linear and angular velocities to start the next time step, after which the simulation continues with only the aerodynamic equations of motion.

Initial Conditions and Other Assumptions

To simulate throw trajectories, I also need to supply initial conditions. In this case, we need the release point and release velocity. I assume a release point x = 0, and a release height y = 6 feet. The initial velocity vector is specified with release speed and release angle from horizontal. For this study, I chose to simulate trajectories for 100 different release speeds between 70 mph and 105 mph, and 100 different release angles from 0 and 30 degrees. In total, therefore, I simulated 10,000 unique release speed and release angle combinations. I ran these simulations with 0.0005 second time steps, which is much finer than necessary to accurately estimate the smooth path of the ball in flight, but allows a very accurate determination of the bounce point. In a further refinement, a numerical integration scheme with adaptive step size, with higher resolution as the ball approaches the ground, would certainly improve calculation time.

Since my goal is to compare air throws and bounce throws from the same distance and release speed, I must perform some additional post-processing on the simulated trajectories. To calculate the target distance for each point, I define a target height, or catch height, for both the air throw and the bounce throw. In principle these could be assumed to be the same, but for my primary analysis, I assumed a target height of two feet for the air throw and one foot for the bounce throw. I will discuss this in more detail below, but most simply, these values represent nearly ideal catch heights for the receiver to tag a sliding runner.

With these catch points established, I run through each trajectory and find the point at which the ball reaches the target heights before and after the bounce, and record the distance and total flight time at which the target heights are achieved. In this way, I transform my dataset from trajectories at different release speed and release angle combinations to flight times for both bounce and air throws at different speeds and target distances. Note that the bounce target height is not necessarily reached for every throw, especially those at lower release angles, so I will not have the same number of valid points in my dataset of bounce throws as for that of the air throws.

Figure 5. Example trajectory comparison for 87.3 mph release speed from 203 feet.

An example to illustrate this approach is shown in Figure 5. I have selected the air throw and bounce throw at a 87.3 mph release speed that are found to achieve their respective target heights at 203 feet from release. The height at which each throw intersects the vertical line at the target distance is the assumed two feet for the air throw and one foot for the bounce throw. Thin dashed lines connect points of equal flight time between the two trajectories. Before the bounce, the equal time contour has a negative tilt, which indicates that the bounce throw has covered more horizontal distance in the same time. After the bounce just before the target, the equal time contour just barely turns over to a positive tilt, showing that the air throw was ultimately faster to its target than the bounce throw for these conditions.

Results

Figure 6. Total flight time to target for air throws and bounce throws.

When total time to target is represented as color map as a function of release speed and target distance, we can confirm that, for a given release speed, it will take longer for the ball to reach its target the farther away it is, and for a given target distance, the ball will arrive more quickly the faster it is thrown. Less obvious, though, is whether the bounce throw or air throw is faster for any given speed/distance combination. To more easily visualize this, I take the speed/distance combinations for which a valid bounce throw was found, and subtract the time to target from that of the air throw at the same speed/distance, interpolating when an exact match is not found.

Figure 7. Time to target difference between air throw and bounce throw.

In Figure 7, Combinations of speed and distance for which the bounce throw is quicker than the air throw are shown in red, while blue represents those combinations where the air throw is quicker. The white stripe through the middle represents the locus on which the air throw and bounce throw are at parity. Having included a wider range of possible throw conditions, we observe a clearer picture of the trend that emerged from the original infield throw study. For a given release speed, the bounce throw becomes the optimal choice for longer throws. From a given distance, air throws are quicker when a faster release speed can be achieved.

I believe all of this makes intuitive sense, but we are also able to see the magnitude of the difference between the two throw types. In general, the difference is less than 50 milliseconds (ms), which seems very small. Consider, however, that the fastest runners achieve a top speed of around 30 ft/s according to Statcast Sprint Speed, and a 50 ms difference translates to 1.5 feet of distance covered, which could easily be the difference between an out and a run.

Returning to Ramon Laureano’s throw, Statcast reported that it was released at 92.1 mph from 321 feet. Given the relatively high catch height, and assuming around a 2200 rpm backspin, I estimate that the release angle must have been around 15 degrees, and the total flight time was about 3.5 seconds. From Figure 7, we can deduce that a bounce throw would almost certainly have been quicker to first base. Furthermore, an air throw with a lower catch point would also have been quicker, but the runner returning to first may have presented a significant obstacle to any lower arrival. The runner was out, and the lateral accuracy was perhaps the most impressive aspect of Laureano’s throw anyway. Other similar practical considerations are discussed below.

Comments

Figure 8. Time differences for lower COR, higher friction (left) and higher COR, lower friction (right).

The conclusions above must be sensitive to the physical parameters of the bounce model, vertical COR and the coefficient of friction. Figure 8 shows the time to target differences for two additional simulation sets. On the left, the vertical COR was decreased by 10 percent and the coefficient of friction increased by 10 percent from nominal, which represents a bounce in which more energy is lost. On the right, the vertical COR is increased by 10 percent and the coefficient of friction decreased by 10 percent. The parity line between bounce and air throws moves between the conditions, with the bounce throw favored for more conditions when the ground is bouncier and has lower friction. Overall, though, the picture remains qualitatively the same. A similar picture can be shown for changing the target heights. If the air throw target height is reduced to be the same one foot as the bounce throw, the air throw becomes more favorable over a larger range of speeds and distances.

So far, all bounce throws have been what I might call “short bounces.” As in Figure 5, the catch point is assumed to be as the ball is on the way back up to the target height after it hits the ground. However, if the bounce reaches and exceeds this target height, it will also intersect the target height again as it is on the way back down, which I will call a “long bounce.” An example of this is shown in Figure 9.

Figure 9. Long bounce vs. air throw example trajectory.

As it turns out, the long bounce allows a much lower trajectory than the air throw for certain distances and speeds, so although the ball is travelling at a lower speed for longer, it can still reach its target sooner. The time difference between air throws and long bounce throws is shown in Figure 10.

Figure 10. Time to target differences between air throws and long bounce throws.

The long bounce throw becomes favorable for very long throws only at a given release speed, but it becomes the optimal choice, even over short bounce throws from these distances. Note the time difference scale goes to 100 ms in Figure 10. A nearly perfect example of the long bounce was a throw from Aaron Judge to nab Christian Vazquez at second earlier this season. I estimate that Judge released the ball about 235 feet from second base, so unless the throw was less than 70 mph, a short bounce may have actually been better!

Conclusion

It is a relatively simple question: Is it better to bounce a throw from the outfield than to reach a target on the fly? Combining the physical models of aerodynamics and oblique bounces allows us to calculate relevant trajectories, but we are forced to make many assumptions to test the question. We can make reasonable assumptions about the release and catch points, and the physical parameters of the aerodynamic and bounce models.

In the end, we conclude that for longer distance throws, especially at slower release speeds, it may indeed be quicker to bounce a throw than to reach the target in the air, based on the shorter possible path length for the bounce throw. We have also implicitly assumed that a fielder can choose a bounce throw or air throw from a given location, and will release the ball at the same speed in either case. To assume anything different would greatly complicate the analysis, but it may ultimately be necessary to tell the whole story.

It seems possible that lower release angles might allow the fielder to accelerate the ball longer in his throwing motion, thus releasing at a higher speed for a bounce throw than an air throw. More data on real-life throws would be needed to validate this empirically. It seems safe to say this effect would tend to favor the bounce throw over a larger range set of conditions, but not alter the qualitative conclusions of the present simpler analysis. More data from Statcast or other more specific experiments could also be used to validate the assumed bounce parameters, and populate a more realistic range of bounce parameters across different types of surfaces in different major league parks.

Of course, there are other reasons why a fielder may not want to bounce a throw. One chalk line or stray pebble could make these tidy trajectory predictions totally moot, and send the catcher sprawling to capture an otherwise accurate throw. The likelihood of a clean catch almost certainly goes down for a bounce throw, which must enter into the determination of the optimal choice in a given situation. For a situation in which the time advantage of a bounce throw is marginal, the extra risk may not be worth it. But for a case where the air throw would simply be too late, it almost certainly would be.

As an astute commenter in my article last year pointed out, aiming lower reduces the likelihood of an “air mail.” Assuming the thrower has some variance around his intended release angle, a throw aimed at the receiver’s foot may bounce, or may reach him on the fly, but is less likely to miss him entirely. This tactical decision is almost entirely independent of any possible time benefit. Mix other factors like wind into the calculus, and a complete decision in real time seems far-fetched. Therefore I believe there is value in developing heuristics based on the simplest possible assumptions presented here.

The exact benefit to improving decision making in these types of plays in also unclear. I don’t know how many potential outfield assist plays exist on the margins where an extra 50-100 ms or a few feet would make a difference. This could perhaps be studied as well, and certainly must be if any in-game value is to be assigned to the apparent benefit of bounce throws. For now, though, I am happy to have obtained a fairly clean result from simple physics, and to have developed tools for trajectory calculation that can shed some light on the amazing plays we see every day in baseball.

References and Resources


Andrew is a research engineer from Waltham, Massachusetts. He has contributed to the FanGraphs Community blog, presented at Saberseminar, and appeared as an analytical correspondent on Japanese television. He can be found on Twitter @ADominijanni, where he'll happily talk science, sports, beer, and dogs.
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wroehl
Member
wroehl

How much of the linear relationship between spin and speed illustrated in Figure 3 is due to that single outlier at speed 80, spin 2000? Does removing that point make a substantial difference? I ask because outliers on graphs that are otherwise a globular cloud of points are often both influential and atypical of the overall process in question.

Brad Johnson
Member

I can hand you one data point. In my prime, I threw 75 mph from SS, 86 mph from the mound, and 92 mph in a hand acceleration drill where we effective pitched to a target 10 feet away. I’m pretty sure the hypothesis that lower release points equals greater velocity is a sound one.

MikeS
Member
MikeS

A bounce throw can potentially be cut off by an infielder in the middle of the diamond. An air throw is far less likely to be cut off and will be easier for a runner to read. When the hitter rounds first and see the throw, he knows immediately he can take second uncontested if the outfielder is attempting to reach home on the fly.

DukeCT
Member
DukeCT
What a great article. Lots of incisive comments too. One unexamined assumption in the piece is that a bounce throw is harder to catch. As I was taught, and in turn taught to my teams, the bounce throw is easier to catch and apply in a tag. Part of this is explained by the circular error probability discussion. Also, from a human factors standpoint, the receiver looking down for the ball will have a better peripheral sense of the play and be more prepared to make the tag. Also, a long hop is definitely easier to catch than a short… Read more »
Balk off
Member
Balk off

I’ve always wondered what the point of diminishing return is on missing the cutoff man and not having a throw relayed. How hard does a throw have to be released to overcome its lost velocity over the last 30-40% of its distance, compared to a relayed throw…at what point is the time added from the relay worth it? Or not?