The psychophysics of bouncing: Perceptual constraints, physical constraints, animacy, and phenomenal causality

Michele Vicovaro; Loris Brunello; Giulia Parovel

doi:10.1371/journal.pone.0285448

Abstract

In the present study we broadly explored the perception of physical and animated motion in bouncing-like scenarios through four experiments. In the first experiment, participants were asked to categorize bouncing-like displays as physical bounce, animated motion, or other. Several parameters of the animations were manipulated, that is, the simulated coefficient of restitution, the value of simulated gravitational acceleration, the motion pattern (uniform acceleration/deceleration or constant speed) and the number of bouncing cycles. In the second experiment, a variable delay at the moment of the collision between the bouncing object and the bouncing surface was introduced. Main results show that, although observers appear to have realistic representations of physical constraints like energy conservation and gravitational acceleration/deceleration, the amount of visual information available in the scene has a strong modulation effect on the extent to which they rely on these representations. A coefficient of restitution >1 was a crucial cue to animacy in displays showing three bouncing cycles, but not in displays showing one bouncing cycle. Additionally, bouncing impressions appear to be driven by perceptual constraints that are unrelated to the physical realism of the scene, like preference for simulated gravitational attraction smaller than g and perceived temporal contiguity between the different phases of bouncing. In the third experiment, the visible opaque bouncing surface was removed from the scene, and the results showed that this did not have any substantial effect on the resulting impressions of physical bounce or animated motion, suggesting that the visual system can fill-in the scene with the missing element. The fourth experiment explored visual impressions of causality in bouncing scenarios. At odds with claims of current causal perception theories, results indicate that a passive object can be perceived as the direct cause of the motion behavior of an active object.

Citation: Vicovaro M, Brunello L, Parovel G (2023) The psychophysics of bouncing: Perceptual constraints, physical constraints, animacy, and phenomenal causality. PLoS ONE 18(8): e0285448. https://doi.org/10.1371/journal.pone.0285448

Editor: Robin Baurès, Universite Toulouse III - Paul Sabatier, FRANCE

Received: December 2, 2022; Accepted: April 23, 2023; Published: August 18, 2023

Copyright: © 2023 Vicovaro et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: Data and research material are available on OSF at the following link: osf.io/xz8t3.

Funding: This publication was made possible thanks to the specific contribution of the University of Siena (Italy) for Open Access support and the PSR funding of the Department of Social, Political and Cognitive Sciences (DISPOC), University of Siena (Italy). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Then he squatted down, held the ball about a half-inch from the floor, dropped it.

It bounced, naturally enough. Then it bounced again. And again. Only this was not natural, for on

the second bounce the ball went higher in the air than on the first, and on the third bounce higher

still. After a half minute, my eyes were bugging out and the little ball was bouncing four feet in the

air and going higher each time.

I grabbed my glass. “What the hell!” I said.

(W. Tevis, The Big Bounce, 1958)

If we are asked to represent the typical behavior of a bouncing ball, we would probably imagine a ball falling down, bouncing against a surface, and then moving back off repeatedly with a decreasing speed and a lower peak height after each bounce, until stillness. Instead, we would be really surprised, or even amused, if we saw a ball increasing its speed and peak height after each bounce, as if it were jumping (see the incipit, and see this video at: https://youtu.be/oQs5WJLUFEM). Apparently, we do not need knowledge of energy conservation principles to perceive a bounce or a jump, like we do not need knowledge of the physics of light to perceive colors.

From a perceptual standpoint, bouncing can be defined as an event, that is, as a sequence of motions with a definite, meaningful perceptual structure. A prominent feature of visually perceived events is phenomenal causality, that is, the impression that the motions of the objects in the scene are causally related [1]. For instance, in Michotte’s [2] launching effect, observers are presented with two horizontally aligned squares, and at a point in time one square (A) starts moving towards the other (B). When A touches B, B starts moving with the same velocity as A, whilst A came to a stop. This simple configuration leads to the vivid visual impression that A launches or kicks B, that is, that the motion of B is caused by the collision with A. The phenomenon is impervious to observers’ explicit knowledge and expectations, which provides support to its perceptual (rather than post-perceptual) origins [3–6]. As another example of visual event, White and Milne [7] presented the participants with simple 2D animations showing an object (A) moving towards a set of initially stationary elements that, upon contact with A, started moving in different directions. Under appropriate conditions, this stimulus configuration led to the vivid impressions of enforced disintegration or bursting. Other examples of visually perceived events include triggering, entraining, expulsion [2], braking [8], penetration [9], pulling [10], generative transmission [11], intentional reaction [12], and shattering [13].

According to Gestalt-theoretic accounts, visual perception of events depends on specific perceptual principles that are unrelated to physical laws, such as motion ampliation [2], coincidence avoidance [14, 15], and grouping [16, 17]. Within the Gestalt-theoretic perspective, various authors have upheld the hypothesis that the laws underlying the visual perception of events are independent of physical laws and of the observer’s prior experience with the physical environment (see also [2, 6, 18, 19]). In this perspective, physical realism would be neither necessary nor sufficient for the visual perception of events.

Other authors have instead emphasized the possible role of past experience in shaping the visual perception of events and the related visual impressions of causality. For instance, according to White’s [20, 21] schema-matching model, information in the visual stimulus would be matched to schemas acquired through everyday perceptual-motor experience with physical objects. Subjective motor experiences of forces do not faithfully represent the corresponding physical forces, which explains why the laws underlying visually perceived events may significantly depart from the predictions of Newtonian physics (e.g., at odds with Newton’s third law, forces generated by collisions are perceived to be asymmetrical [20, 22]). In a similar vein, Hubbard [23] has suggested that causality would be perceived when the visual stimulus matches an impetus transmission heuristic, by which an agent (e.g., A in the launching effect) is seen to impart or transmit an impetus to a patient (e.g., B; see also [24]). The impetus transmission heuristic would be acquired from motor experience with physical forces (see also [25]). Two noteworthy characteristics of the impetus transmission heuristic are that the impetus is conserved (i.e., the impetus of the patient cannot be greater than the impetus transmitted to it by the agent) and that impetus dissipates, leading to a gradual slowdown of the patient.

Models based on schema-matching and on impetus transmission heuristic emphasize the discrepancies between the laws underlying the visual perception of events and the corresponding Newtonian laws. By contrast, according to the noisy Newton model, the perception and the interpretation of physical events would be based on internalized physical laws [26–30]. According to this model, events perception can be reconciled with the predictions from Newtonian mechanics, if uncertainty due to sensory noise is taken into account. However, Kominsky et al. [31] argued that perceptual constraints may impose limits on the number of physical constraints that can actually be internalized, and on the precision with which they are represented. For instance, in launching-like scenarios, representations of the range of physically possible speed ratios between A and B would be imprecise due to the limited sensitivity of the visual system to speed variations.

The first aim of the present work was to explore the laws underlying the visual perception of physical motion and animated motion in bouncing-like scenarios, and to provide a systematic comparison between these laws and the corresponding physical laws. Specifically, Experiments 1 and 2 explored how the visual perception of physical motion and animated motion are affected by a variety of physical parameters (i.e., coefficient of restitution, motion pattern, simulated value of gravitational acceleration, number of bouncing cycles, presence of a time delay at the moment of the impact between the bouncing object and the bouncing surface). The results can provide novel insights about the possible interplay between physical and perceptual constraints in shaping the visual perception of events, and more in general they can inform the debate around current theories of causal perception. In Experiment 3, we tested the necessity of an opaque visible bouncing surface for the visual perception of physical motion and animated motion in bouncing-like scenarios. Lastly, Experiment 4 explored if and how visual impressions of physical motion and animated motion are related to visual impressions of causality.

The physics of bouncing

We focus on the behavior of an object that falls vertically on a rigid horizontal surface [32]. For simplicity, hereafter we refer to a bouncing cycle as a sequence of events formed by 1) a falling phase, in which the object falls downwards under the effects of gravitational acceleration; 2) a collision against a rigid bouncing surface; 3) a climbing-back phase in which the objects move upwards (again, under the effects of gravitational acceleration) until stillness.

In the falling phase, the object accelerates downwards due to the effects of gravity. Ignoring the possible effects of air resistance, the object’s acceleration is g = 9.80665 ≈ 9.81 m/s², and its velocity just before the collision with the surface (v₀) is given by: (1) where h₀ is the object’s initial height from the surface (i.e., the falling height). The duration t₀ of the falling phase is (2)

Upon contact with the surface, the object compresses and then expands, bouncing back off the surface and moving upwards. The object’s collision behavior depends on a complex interaction between its material properties, the material properties of the surface, the object’s shape and its possible spin [33]. Ignoring these complexities, the object’s bouncing behavior is well described by Newton’s Law of Restitution, according to which (3) where v₁ is the object’s post-collision velocity and C is the coefficient of restitution. Here the minus sign represents the inversion of the motion direction after the collision with the surface. In terms of energy conservation and dissipation, upon the collision with the surface, the initial kinetic energy of the falling object is converted to elastic potential energy, which is then converted back to kinetic energy. Because of energy conservation principles, the kinetic energy after the collision (i.e., 0.5mv₁²) cannot be greater than the kinetic energy before the collision (i.e., 0.5mv₀²), which means that C cannot be greater than 1. In the energy transformation processes, some energy is always dissipated as heat, therefore C is typically smaller than 1. Moreover, C cannot be smaller than 0, otherwise the post-collision velocity would have the same direction as the pre-collision velocity (i.e., the object would penetrate inside the surface). All this implies 0 ≤ C ≤ 1, that is, 0 ≤ |v₁| ≤ v₀.

The coefficient of restitution C constitutes a link between the material properties of the bouncing object and its motion pattern [34]. Relatively inelastic objects such as balls made of plasticine or clay are associated with small values of C, and tend to stick to the surface after the collision (i.e., high energy dissipation; low post-collision speed and peak height). On the contrary, relatively elastic objects such as tennis balls and superballs are associated with high values of C, and tend to bounce high after the collision with the surface (i.e., low energy dissipation; high post-collision speed and peak height).

In the climbing-back phase, the object decelerates until it reaches a peak height h₁ given by (4)

The duration t₁ of the climbing-back phase is (5)

After the first bounce, a new bounce starts, and h₁ is the starting position of the object in the second falling phase. A variable number of bounces may occur, until the object remains still on the surface.

The intuitive physics of bouncing

In the case of bouncing, the motion pattern of the bouncing object is uniquely related to parameter C, and therefore to the elasticity of the bouncing object itself. More specifically, C is specified by the ratio of the velocities before and after the collision, by the ratio of the peak heights, and by the ratio of the periods of successive bounces: (6) where τ₀ is the period of the first bounce (i.e., t₀+t₁) and τ₁ is the period of the second bounce [35].

Previous studies on bouncing perception have mainly focused on observers’ ability to perceive the elasticity of the bouncing object from the bounce kinematics. Warren et al. [35] showed that participants could accurately estimate the elasticity of a simulated bouncing object (i.e., a simulated disk falling vertically downwards and bouncing various time on a flat surface), mainly relying on the ratio of peak heights. Similar results have also been reported by Nusseck et al. [36] and Paulun and Fleming [37] in more realistic bouncing scenarios.

Elasticity ratings can be used to explore participants’ intuitive knowledge of the relationship between the object’s elasticity and its kinematics, but they do not provide information about whether observers can discriminate between physically possible and physically impossible bounces. However, in their study, Paulun and Fleming [37] also asked participants to rate the typicality of the motion of virtual bouncing cubes, and reported a negative relationship between elasticity and typicality. In other words, there would be a realistic perceptual prior for relatively inelastic objects in the real world.

Twardy and Bingham [38] presented participants with a simulated small-scale 3D environment showing a sphere falling to the ground from high above with a parabolic trajectory and bouncing against the simulated ground multiple times. Results showed that the naturalness ratings were relatively high for C ≤ 1 (with a peak for C = 0.9), whereas, consistently with physics, motions characterized by C > 1 were judged as unnatural. In addition, participants provided low naturalness ratings to animations showing decreasing gravity, whereas they were relatively insensitive to gravity increases. These results indicate that conservation violations are perceptually unnatural.

Previous studies on the intuitive physics of bouncing have used relatively realistic virtual scenarios showing an object moving in a 3D environment [37, 38]. However, converging evidence indicates that events can be unambiguously perceived even in the case of schematic 2D animations showing simple shapes moving on a uniform background (for a review see [1]). As a demonstration of their being deeply rooted in early visual processing, studies have shown that visually perceived events can influence visual memory [24, 39] and low-level spatiotemporal properties of the scene, such as distance [40, 41], trajectory [42], speed [43, 44], and timing [45].

From physical to animated motion

So far, we have focused on visual impressions involving the motion of inanimate objects. However, under appropriate conditions, animations may give rise to vivid impressions of animacy. For instance, when the stimuli that give rise to the launching effect are modified so that the post-collision velocity of B is more than two times as large as the pre-collision velocity of A, the launching effect leaves place to the so-called triggering effect [2, 18]. In this case, the observers have the impression that the motion of B is triggered by the contact with A, however the motion of B is perceived as active, that is self-generated by B and not passively caused by the collision with A as in the launching effect. Moreover, when B starts moving before the collision with A and faster than it, observers have the impression of an intentional reaction in which B is seen as self-propelling and intentionally escaping at the arrival of A [12]. In these cases, a hidden inner force is attributed to B, that makes it move independently of the force transmitted to it by A.

Since these pioneering works, many authors have argued that the perception of animacy is triggered by the manipulation of spatiotemporal contingencies between two or more moving objects, or between a moving object and other contextual elements (e.g., [46–50]). A general factor that triggers animacy impressions would be the apparent violations of Newtonian principles (e.g., [51, 52]). Other simpler cues to animacy are self-propulsion [53, 54] and non-rigid, rhythmic expansion-contraction motion [2, 55, 56]. According to several authors, phenomena like causality and animacy, despite the seemingly higher-level impressions they prompt, are vivid, irresistible and dependent entirely on basic low-level display parameters (e.g., [4, 5, 19]).

Bingham et al. [57] have provided some empirical support to the hypothesis that bouncing animations can sometimes leave place to perceived animacy, by showing that observers could discriminate real physical bounces from hand-driven bounces with the same period. In our present research we provide a systematic exploration of the conditions by which the visual impression of bouncing may give way to impressions of animate behaviors such as jumping. We adopt a rather neutral approach to animacy perception, leaving open the possibility that jumping may not be the only type of animated motion that emerges in bouncing-like scenarios (e.g., the animated motion of an object that goes back-and-forth from a surface).

Experiment 1

The aim of this first experiment was to provide a broad exploration of animation parameters that lead to the perception of physical motion or animated motion in bouncing-like scenarios. Participants were presented with simple 2D animations such as the one shown in Fig 1, and were asked to indicate if the animation showed a physical bounce, an animated motion, or neither of these two. Four parameters were manipulated. First, the value of C, which is a crucial parameter in the physics of bouncing. If observers are sensitive to the violation of conservation laws, as it was suggested in previous studies [37, 38], then we hypothesize that physical bounce impressions should emerge for simulated values of C smaller than 1, whereas values greater than 1 would be consistent with the perception of animated motion.

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Fig 1. Example of the animations used in Experiment 1 (four frames).

The blue arrow was added in this figure for illustrative purposes, to show the motion direction of the disk.

https://doi.org/10.1371/journal.pone.0285448.g001

The second manipulated parameter was the motion pattern, as the falling object could move with uniform velocity or uniform acceleration (i.e., uniform acceleration in the falling phase and uniform deceleration in the climbing-back phase). This factor has not been manipulated in previous studies on the perception of bouncing, however Vicovaro et al. [58] showed that physically implausible vertical falls characterized by uniform velocity were perceived approximately as natural as accelerated vertical falls that were consistent with terrestrial gravitational acceleration. We wanted to explore whether, also in the case of bouncing, observers are relatively insensitive to the presence or absence of simulated gravitational acceleration.

The third manipulated factor was the simulated value of gravitational acceleration (a), which could correspond to terrestrial gravitational acceleration or to values smaller than gravitational acceleration. Recent studies have shown that objects are believed to fall vertically downward much more slowly than how they actually fall under the influence of terrestrial gravitational acceleration [59–61]. Based on this prior evidence, we wonder if the perception of physical bounce is more likely under micro-gravity conditions.

Lastly, we also manipulated the number of bouncing cycles, which could be one or three.

Methods

Transparency and openness.

In this and the following experiments, we report how we determined our sample size, all data exclusions (if any), all manipulations, and all measures in the study, and we follow JARS [62]. Data, samples of stimuli, supplementary figures and supplementary tables are available on OSF at the following link:

https://osf.io/xz8t3/?view_only=02450de951bd48e89e8313a6dff5be52. Data were analyzed using R, version 4.0.4 [63]. This study’s design and its analysis were not pre-registered.

Sample size.

This and the following experiments are exploratory in nature, therefore the sample size is not based on the estimation of a definite expected effect size. We decided to test 30 participants in each experiment, a sample size similar to that of previous studies on visual perception of events [7, 9–11, 13, 17, 64] and larger than the sample size of previous studies on the intuitive physics of bouncing object, in which N < 15 [37, 38].

Participants.

Thirty graduate or undergraduate students at the University of Padova participated in the experiment on a voluntary basis. They were aged from 20 to 31 years (M = 23.63 years, 95% CI [22.59, 24.67]), 18 were women and 12 were men. All participants were naïve to the purpose of the experiment, and gave written informed consent according to the Declaration of Helsinki prior to their inclusion in the experiment. They all reported to have normal or corrected-to-normal visual acuity and were rewarded with 10 €. All procedures used in this and the following experiments were approved by the Ethics Committee for Research in the Human and Social Sciences—CAREUS of the University of Siena (protocol number 194793).

Stimuli and apparatus.

The stimuli were generated with PsychoPy3 [65] and were presented on a 43.2 cm × 27 cm LCD screen (refresh rate 60 Hz). Participants sat at a distance of about 50 cm from the screen, the background of which was grey [RGB(.506,.506,.506)]. At the beginning of each animation, a black circle (diameter = 1 cm; hereafter disk) appeared with its center 4 cm below the center of the screen. A black, thin horizontal rectangle (height = 0.5 cm, length equal to the screen width) also appeared adjacent to the bottom edge of the screen, and remained visible for the whole duration of the animation. This rectangle (hereafter surface) was meant to represent a flat horizontal surface against which the disk would bounce at the end of the fall. After 500 ms, the disk started moving vertically downwards according to the parameters described below. A schematic depiction of a stimulus animation is shown in Fig 1.

For convenience, each bouncing cycle can be arbitrarily divided into three phases: the falling phase, the collision, and the climbing-back phase. The falling phase lasts from when the disk starts moving downwards to when it collides with the surface. In this experiment, the motion of the disk during the falling phase was given by the combination of factors a and motion pattern. Parameter a could take three different values, that is g (9.81 m/s²), g/4 (2.45 m/s²) or g/16 (0.61 m/s²). For reference, on the Moon the gravitational acceleration is about g/6, on Mars it is about g/2.6. The motion pattern could be uniform acceleration or uniform velocity. In the case of uniform acceleration, the disk started moving from rest with acceleration a, until it reached a final velocity v₀ = √(2ah₀) just before the collision with the surface (h₀ is the initial distance of the disk from the surface, 8.5 cm). As for uniform velocity, the disk moved at a constant velocity √(2ah₀)/2. That is, the velocity of the disk corresponded to the average velocity of a disk falling from height h₀ with acceleration a.

The collision is the moment of the impact between the disk and the surface. The crucial parameter is the simulated coefficient of restitution C, which determines the disk’s post-bounce behavior according to equation v₁ = -(v₀ × C), where v₁ is the velocity of the disk immediately after the collision. Values of C greater than one would be physically implausible because they would imply a violation of energy conservation. In this experiment, C could take nine different values (i.e., 0.25, 0.5, 0.75, 0.875, 1, 1.125, 1.25, 1.5, 1.75). There was no time delay between the moment the front edge of the disk touched the bouncing surface and the beginning of the climbing-back phase. Specifically, the disk made contact with the surface for only one frame.

The climbing-back phase lasts from immediately after the collision to when the disk reaches a state of stillness. Similar to the falling phase, the motion of the disk was determined by the combination of factors a and motion pattern. In the case of uniform acceleration, the disk moved with a negative acceleration–a until coming to a stop at height h₁ = v₁²/2a from the surface. As for uniform velocity, the disk moved at a constant velocity -√(2ah₁)/2. That is, the velocity of the disk corresponded to the average velocity of a disk climbing back from the surface to height h₁ with acceleration -a.

The animations could show either one bouncing cycle or three bouncing cycles. In the case of one bouncing cycle, the animation was stopped immediately after the end of the first climbing-back phase. In the case of the three bouncing cycles, the sequence of falling phase, collision, climbing-back phase was repeated three times (i.e., the animation was stopped immediately after the third climbing-back phase). The falling height at the beginning of each cycle corresponded to the final height of the disk at the end of the previous cycle, as it would be for a real physical bounce.

One-hundred and eight animations were built according to the following factorial design: 2 number of bouncing cycles (1 or 3) × 3 a (0.61 m/s², 2.45 m/s², 9.81 m/s²) × 2 motion pattern (uniform acceleration or uniform velocity) × 9 C (0.25, 0.5, 0.75, 0.875, 1, 1.125, 1.25, 1.5, 1.75). In addition, 12 variable a animations and 12 variable C animations were also created, all showing three bouncing cycles. In the variable a animations, a different value of a was used for each bouncing cycle, as if the gravitational attraction varied across bouncing cycles. In these animations, C was kept constant at 0.75. Considering the three bouncing cycles, there were six possible permutations of the three levels of a (e.g., 9.81/2.45/0.61 m/s², 9.81/0.61/2.45 m/s², etc.). For each of these permutations, two variable a animations were created, one showing uniform acceleration and the other showing the corresponding uniform velocity. Symmetrically, in the variable C animations a different value of C was used for each bouncing cycle, whereas a was kept fixed at 2.45 m/s². Three values of C were selected arbitrarily (i.e., 0.5, 1.0, 1.5), which results again in six possible permutations. There were two animations for each permutation, one showing uniformly accelerated motion and the other showing the corresponding uniform velocity motion.

To sum up, there were a total of 132 animations (i.e., 108 animations from the factorial design + 12 variable a animations + 12 variable C animations). Each animation was presented three times in a random order, for a total of 396 experimental trials. A sample of the stimuli is available on OSF.

Procedure.

Instructions that were readable on the screen informed the participants that they would be presented with bouncing animations, and that their task was to classify each animation into one of three possible categories (i.e., a 3-AFC task). More precisely, participants were presented with the following instructions: “a) physical bounce (key 1): the object bounces plausibly against the surface, as if it were an inanimate object (e.g., a ball); b) animated motion (key 2): the object moves like a living being with its own internal force; c) other (key 3): the animation does not represent a physical bounce or an animated motion.” Instructions also invited the participants to trust their first impression. Reviewing the animations was not permitted, and only the numbered keys on the top left of the keyboard could be used to provide the response. We employed a 3-AFC task instead of a 2-AFC task because our initial examination of the stimuli revealed some motions that were not easily described as physical bounce or animated motion. By using a 3-AFC task, we aimed to avoid that participants could classify these ambiguous stimuli as either physical bounce or animated motion, which would have resulted in a greater diversity of visual impressions within the two primary response categories. Instead of using the 3-AFC task, other options could be considered such as using separate rating scales for each of the three response categories, or presenting pairs of stimuli side-by-side and asking participants to choose the stimulus that best represents each category. These methods may provide more detailed information compared to the 3-AFC, especially the rating method. However, it is important to consider that these alternatives may require a longer experimental time and result in a reduction in the number of stimuli used. This may not be ideal since our primary goal was to broadly explore the relationship between the physical parameters and resulting visual impressions.

Each experimental trial started with a 500-ms blank screen, followed by one randomly selected animation. Immediately after the animation ended, a response screen was presented to remind the participant of the correct key category associations. There were no feedback or time limits for the response. After a response was provided, a new trial started. A custom-duration pause was presented after every 27 experimental trials.

One-hundred sixty-two one bouncing cycle animations (54 animations × 3 repetitions) and 234 three bouncing cycles animations (78 animations × 3 repetitions) were presented in two different experimental sessions that took place at least 24 hours apart, in counterbalanced order. Within each session, the animations showing uniform acceleration and those showing uniform velocity were presented in different blocks, the order of which was counterbalanced across participants. Within each block, the animations were presented in a random order. Before starting each block, participants were presented with randomly selected practice trials to familiarize them with the task (i.e., five practice trials for the one bouncing cycle animations and nine practice trials for the three bouncing cycles animations).

Results and discussion

Figs 2 and 3 show the mean probabilities, averaged across the participants, of physical bounce responses (blue curves), animated motion responses (red curves), and other responses (grey curves) as a function of C, separately for uniform acceleration and uniform velocity, and separately for the three values of a. Figs 2 and 3 refer respectively to one bouncing cycle and three bouncing cycles animations (the results for the 12 varying a and the 12 varying C animations are not represented in these graphs). An alternative representation of the results is provided in Supplementary Fig S1 and S2 on OSF. These figures show, in each cell, the mean probabilities, averaged across the participants, of physical bounce responses (lower left corner) and animated motion responses (upper right corner) for one bouncing cycle (Fig S1), three bouncing cycles (Fig S2), and varying a/varying C animations (Fig S3). Blue cells indicate animations associated with clear impressions of physical bounce, whereas red cells indicate animations associated with clear impressions of animated motion. An animation was classified as an instance of clear physical bounce impression (animated motion impression) if the probability of physical bounce response (animated motion response) exceeded .50. Grey cells correspond to animations that did not elicit clear impressions of physical bounce or animated motion, but that at the same time were associated with a low mean probability of other responses (i.e., smaller than .33; please note that the probability of other responses is one minus the sum of the other two probabilities). Therefore, grey cells represent somewhat ambiguous animations in between physical bounce and animated motion.

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Fig 2. Observed probability of physical bounce responses (blue curves), animated motion responses (red curves), and other responses (grey curves) for the one bouncing cycle animations of Experiment 1.

The shaded area around each curve represents the 95% confidence interval for binomial variables. The horizontal dashed line highlights the .50 response probability threshold.

https://doi.org/10.1371/journal.pone.0285448.g002

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Fig 3. Observed probability of physical bounce responses (blue curves), animated motion responses (red curves), and other responses (grey curves) for the three bouncing cycle animations of Experiment 1.

The shaded area around each curve represents the 95% confidence interval for binomial variables. The horizontal dashed line highlights the .50 response probability threshold.

https://doi.org/10.1371/journal.pone.0285448.g003

The analyses were conducted on mean response probabilities. In the case of physical bounce, the response was coded as 1 if the participant responded physical bounce and as 0 if the participant responded animated motion or other. A similar approach was used for the coding of the other two response categories. The observed probability of each response category was then analyzed using repeated measures ANOVA with Huynh-Feldt correction for the violation of the sphericity assumption. The use of ANOVA for the analysis of dichotomous data has been subject to criticism [66]. However, recent simulation studies [67, 68] have demonstrated that repeated measures ANOVA, with Huynh-Feldt correction, provides better control over Type I and Type II errors than alternative models, such as generalized mixed-effects logit models.

We first run a 2 (Number of bouncing cycles) × 2 (Motion pattern) × 3 (a) × 9 (C) repeated measures ANOVA on the probability of each response category. These analyses were limited to the data from the 108 animations generated from the main factorial design, whereas the data for the 12 varying a and the 12 varying C animations were analyzed separately. The results are reported on OSF in Supplementary Tables S1 (physical bounce), S2 (animated motion), and S3 (other). The analysis of physical bounce responses revealed a statistically significant four-way interaction. Additionally, there were statistically significant two- and three-way interactions involving the number of bouncing cycles for each of the three response categories. Therefore, to simplify the analysis and discussion of the results, separate ANOVAs were conducted for the one bouncing cycle and the three bouncing cycles conditions. The results are reported in Table 1 (physical bounce) and in Table 2 (animated motion), whereas the results for other responses are reported in Supplementary Table S4 on OSF, and they will not be discussed in the main text.

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Table 1. ANOVA results (Experiment 1) for physical bounce responses, separately for one bouncing cycle and three bouncing cycles.

https://doi.org/10.1371/journal.pone.0285448.t001

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Table 2. ANOVA results (Experiment 1) for animated motion responses, separately for one bouncing cycle and three bouncing cycles.

https://doi.org/10.1371/journal.pone.0285448.t002

To facilitate the discussion, we will initially focus on the influence of motion pattern, a and C on physical bounce and animated motion responses, specifically comparing the results for one bouncing cycle and the results for three bouncing cycles. Subsequently, we will address the interaction effects.

Motion pattern.

As shown in Table 1 and by the comparison between the left-side and the right-side panels in Figs 2 and 3, the probability of physical bounce responses was larger in the case of uniform acceleration than in the case of uniform velocity, in the case of one bouncing cycle (M = .49, 95% CI [.31, .66] and M = .37, 95% CI [.19, .54]) as well as in the case of three bouncing cycles (M = .43, 95% CI [.25, .60] and M = .36, 95% CI [.19, .54]). Symmetrically, the probability of animated motion responses was larger in the case of uniform velocity than in the case of uniform acceleration, although the difference reached a statistically significant level in the case of one bouncing cycle (M = .42, 95% CI [.24, .59] and M = .32, 95% CI [.15, .49]) but not in the case of three bouncing cycles (M = .39, 95% CI [.22, .57] and M = .37, 95% CI [.19, .54]).

These results suggest that uniform acceleration/deceleration provided a cue to physical bounce, whereas uniform velocity provided a cue to animated motion. However, motion pattern had a stronger effect in the case of one bouncing cycle (η_G² = .027 for physical bounce responses and η_G² = .017 for animated motion responses) than in the case of three bouncing cycles (η_G² = .008 for physical bounce responses and η_G² = .001 for animated motion responses). Fig S1 shows that, in the case of one bouncing cycle, 14 out of the 16 animations that gave rise to clear physical bounce impressions were characterized by uniform acceleration (87.5%), whereas 4 out of 5 clear animated motion impressions were characterized by uniform velocity (80%). These percentages were comparatively lower in the case of three bouncing cycles (i.e., 57.1% for physical bounce and 46.1% for animated motion, see Fig S2).

a. The probability of physical bounce responses decreased with a (9.81 m/s²: M = .26, 95% CI [.10, .41]; 2.45 m/s²: M = .45, 95% CI [.27, .63]; 0.61 m/s²: M = .52, 95% CI [.34, .70]). No clear differences between one and three bouncing cycles emerged, as also demonstrated by the large p-value and the negligible effect size for the interaction between the number of bouncing cycles and a in the four-way ANOVA (p = .654 and η_G² = < .001, see Supplementary Table S1). As for animated motion impressions, a had a negligible non-significant effect, again with no apparent differences between one and three bouncing cycles (p = .910 and η_G² = < .001 for the interaction between the number of bouncing cycles and a in the first four-way ANOVA, see Supplementary Table S2).

Surprisingly, none of the animations characterized by a = 9.81 m/s² (i.e., the terrestrial gravitational acceleration) gave rise to a clear impression of physical bounce (see Figs 2, 3, S1 and S2). That is, not only physical bounce impressions were favored by simulated micro-gravity, but simulated terrestrial gravitational attraction was actually incompatible with physical bounce impressions. We will return on this important result in the General Discussion.

C. Parameter C had a significant effect on physical bounce responses in the case of one bouncing cycle as well as in the case of three bouncing cycles. The effect size was noticeably larger for three bouncing cycles (η_G² = .104) compared to one bouncing cycle (η_G² = .041). Furthermore, the impact of C on physical bounce responses differed considerably between one bouncing cycle (Fig 2) and three bouncing cycles (Fig 3) at a qualitative level.

In Fig 3 (three bouncing cycles), the blue curves for physical bounce responses were similar in shape to classic psychometric curves. Except when a = 9.81 m/s², relatively large probability values emerged for values of C in the range from 0.25 to 0.75, then probability decreased steeply in the range from 0.75 to 1.5 (see Fig 3C–3F). This suggests that observers were very sensitive to the physical plausibility of C in the case of three bouncing cycles, which is also supported by the fact that clear physical bounce impressions emerged only when C was physically plausible (i.e., C ≤ 1) whereas clear animated motion impressions emerged only when C was physically implausible (i.e., C > 1; see Fig 3 and S2). As shown in Fig 3C–3F, the intersection between physical bounce and animated motion responses occurred in the interval from 0.875 to 1.125 (i.e., close to 1, which is the physical threshold of plausibility of C).

In Fig 2 (one bouncing cycle), the blue curves for physical bounce responses showed a reversed U-shape pattern with an initially increasing trend, a peak for values around 0.875–1.25, and then a decreasing trend. As shown in Fig 2E and S1, with uniform acceleration and a = 0.61 m/s², clear physical bounce impressions emerged for all values of C except 0.25. Therefore, even animations showing extreme violations of energy conservation (e.g., C = 1.75) led to the perception of clear physical bounce. In other words, provided that uniform acceleration was present, almost all values of C elicited a clear physical bounce impression. Additionally, it is noteworthy that, with uniform acceleration and a = 2.45 m/s² or a = 0.61 m/s² (Fig 2C and 2E), physical bounce responses were always more likely than animated motion responses for all values of C. Indeed, there was no intersection between physical bounce responses and animated motion responses in these cases. Surprisingly enough, the only value of C associated with a low probability of physical bounce responses was a physically plausible one (i.e., 0.25). This can be explained by the fact that, in the case of one bouncing cycle with C = 0.25, the climbing-back phase was very short, therefore the animation ended rather abruptly after the first bounce (i.e., the animation was probably too short to provide a clear bouncing impression; this hypothesis is also supported by the high observed probability of other responses).

As for animated motion impressions, they tended to increase with C, in the case of one bouncing cycle as well as in the case of three bouncing cycles (see Table 2, Figs 2 and 3). Again, the effect was clearly more pronounced in the case of three bouncing cycles (η_G² = .139) than in the case of one bouncing cycle (η_G² = .060; see also the significant interaction between the number of bouncing cycles and C in Table S2).

To sum up, the results of these analyses indicate that the visual system is attuned to the physical plausibility of C in the case of three bouncing cycles, but not in the case of one bouncing cycle. It is also worth noting that the opposite tended to be true for the motion pattern. In other words, the number of bouncing cycles modulates the influence of C and motion pattern on the participants’ responses. The theoretical implications of these results will be discussed in more details in the General Discussion.

Interaction effects.

Table 1 highlights that for physical bounce impressions with three bouncing cycles, all two-way interactions and the three-way interaction were found to be statistically significant. In terms of interpreting these interactions, the visual inspection of Fig 3 suggests that the effect of each factor on the probability of physical bounce responses was more pronounced at the level(s) of the other factor(s) that were related to a higher likelihood of physical bounce impressions. For instance, the effect of C was more prominent in the case of uniform acceleration as compared to uniform velocity. Additionally, it was stronger for the two smaller values of a in comparison to the largest value. The interactions observed for physical bounce impressions with one bouncing cycle (Table 1, Fig 2), as well as for the statistically significant interactions seen for animated motion impressions (Table 2, Figs 2 and 3), seem to follow a similar pattern.

Results for the “varying a” and the “varying C” animations.

As shown in Fig S3, the varying a animations were mostly associated with low probabilities of physical bounce responses (M = .24, 95% CI [.16, .32]) and with moderate probabilities of animated motion (M = .37, 95% CI [.25, .48]) and other responses (M = .40, 95% CI [.28, .51]). None of these animations gave rise to clear physical bounce or animated motion impressions.

The varying C animations were mostly associated with low probabilities of physical bounce responses (M = .24, 95% CI [.16, .31]), low probabilities of other responses (M = .31, 95% CI [.21, .41]), and relatively high probabilities of animated motion responses (M = .46, 95% CI [.36, .55]). Only two of these animations gave rise to clear animated motion impressions, however a probability close to .50 was observed for most of these animations. Fig 4 shows that the only animation associated with a relatively low probability of animated motion response was the one in which C decreased across the three bouncing cycles.

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Fig 4.

Observed probability of physical bounce responses (blue curves), animated motion responses (red curves), and other responses (grey curves) for the four animations types of Experiment 2, as a function of the delay. The shaded area around each curve represents the 95% confidence interval for binomial variables. The horizontal dashed line highlights the .50 response probability threshold. Panels A and B: results for one bouncing cycle animations. Panels C and D: results for the three bouncing cycles animations. Panels A and C: results for C = 0.75. Panels B and D: results for C = 1.25.

https://doi.org/10.1371/journal.pone.0285448.g004

Experiment 2

In the second experiment we tested to see if and how a short delay at the moment of the collision would affect visual impressions of physical bounce and animated motion. Michotte [2] showed that observers could perceive a clear launching effect when a 0–70 ms delay was present between the arrival of A and the departure of B. The strength of the launching effect was even improved by a 30–40 ms delay, whereas delays longer than 70 ms tended to progressively weaken and then to disrupt the launching impression (see also [1], for a review of studies on the influence of timing on phenomenal causality). Michotte [2] argued that the reason why short delays do not disrupt the launching effect despite their physical implausibility is that such delays do preserve the perceived temporal continuity of the motions of A and B, which is necessary for the effect to occur.

From a physical viewpoint, the delay between the moment in which a falling object touches the bouncing surface and the subsequent climbing-back phase is typically negligible (e.g., 5 ms for a tennis ball, 0.1 ms for a steel ball [32]). While the delay might be longer for certain objects, like a rubber ball that is not fully inflated, it is noteworthy that the stimuli utilized in our experiments did not display any noticeable deformation upon impact with the bouncing surface. The stimuli resembled rigid steel balls more than soft rubber balls, as seen in the stimuli provided on OSF. Therefore, if the visual system is sensitive to the plausibility of the event’s physical properties, delays as brief as 30 milliseconds should be enough to disrupt the impression of a physical bounce. Alternatively, if the key factor is the perceived temporal continuity between the falling and the climbing-back phase, then the tolerance to delays should be similar to that reported by Michotte [2] in the case of the launching effect (i.e., 0–70 ms).

As for the relationship between temporal delays and animated motion, the behavior of living bodies can be compared to that of a spring, especially when one considers the biomechanics of jumping. The body as a whole shrinks at the moment of the contact with the surface and then expands to bounce off the surface. This shrinking-expansion process plausibly takes a relatively long time, therefore we speculate that relatively long delays between the contact with the surface and the climbing-back phase may favor visual impressions of animated motion (i.e., jumping).