This experiment investigates how the human cognitive system represents dynamic motion. Mechanically, dynamic motion can be characterized by the velocity and the mass of an object. Velocity, a combination of the speed and the direction of motion, is the kinematic feature of the object, whereas the amount of mass that the object has is a measure of the dynamic content of the object.
A well-known theory on human conception of dynamic motion is the Impetus Theory developed by McCloskey (McCloskey, 1983). In one experiment, he gave his subjects a spiral tube as shown in Figure 4. They were told that a metal ball is shot into the tube at one end as indicated by the arrow in the figure; it travels at high speed inside the tube and comes out the other end. Their task was to draw the path of the ball outside the tube. It turned out that 51% of the subjects had drawn paths which are curved, as shown in the figure. McCloskey claimed that the subjects conceived of the ball as carrying impetus, a commonsense concept of force. The subjects who drew curved paths conceived of the ball as carrying a curved impetus, which is responsible for the ball's curved motion.
Similar stimuli were used for the present experiment. The key question was whether subjects' curved responses would differ if, in the stimuli, they were told that balls of different mass were shot into the tube. In other words, if people did have the concept of curved impetus, how exactly would such a conception be affected by or related to the amount of mass assigned to the ball? Would a lead ball traveling inside and coming out of the tube be conceived as curving differently from a ping-pong ball? If so, people's conception of impetus associated to an object in motion would also depend on their conception of mass of the object. Then, can we consider the conception of mass as a component in the conception of impetus? If not, is there any alternative theory that can be used to explain the result?
Mass is not the only factor that might influence people's conception of mechanical motion. In this experiment, I also studied how the speed and the cohesiveness of objects would affect people's responses. Are people cognitively sensitive to the speed as much as they are to the mass of an object in mechanical motion? A lead ball is physically very different from a pile of sand or a gush of water. A lead ball is rigid and cohesive, while sand and water are not. Do people cognitively distinguish such a physical property? Do they conceive differently the mechanical behavior between a lead ball and a pile of sand? If they do, how can we account for the differences in their responses with respect to their mental representations of non-cohesive objects?
Subjects: Two hundred and sixty three undergraduate students participated in this experiment. 58.5% of the subjects were female and 41.5% of them were male. The average age of the entire sample was 20.3 years. Twenty-six percent of these students had taken college physics, forty-three percent of them had taken high school physics, and thirty percent of them had never enrolled in a physics class. The subjects were tested in groups that differed in size, ranging from 3 to 109 students.
Stimuli: Two sets of computer-drawn graphs were the stimuli for this experiment. Set 1 or the Lead-ball stimulus set consists of three stimulus pages and Set 2 or the Ping-pong-ball stimulus set consists of four stimulus pages. Each page was a regular 8.5in×11in size paper. On every page, the same basic drawing is shown which includes a spiral tube, about 630 in its total rotation and 4.5cm in diameter, lying flat on a table surface. The word 'Table' on the bottom left of each page and shading in each graph emphasized to the subjects that the spiral was lying flat on the table and that the subjects were looking down on the table surface when they were looking at the graph. On every page, seventeen white ping-pong balls were also drawn lying on the table above the exit of the spiral tube in a symmetrically fan-shaped curve. The instructions written on the top of every page further emphasized to the subjects that they were looking down on the spiral tube lying flat on the table as they were looking at the graphs.
The basic stimulus material, as described above, were used in every condition of the experiment and thus remained exactly the same in every stimuli. The instructions in the stimuli to the subjects were kept as identical as possible across all conditions except those words that were necessary for specifying the conditions of the experiment. Key words in the instructions were written in boldface. Stimulus material particular to each condition was added to the basic stimuli: a water hose in the Water condition, a pile of sand in the Sand condition, the word 'Fast' in the Fast condition, the word 'Slow' in the Slow condition. Note, the same ball (in color and size) was drawn in the Lead-ball stimuli and the Ping-pong stimuli under both Fast and Slow conditions, i.e., the specifications of the Lead-ball condition and the Ping-pong-ball condition were done purely through the verbal instructions on the stimulus pages.
Procedure: Prior to the experiment, a large number (about three hundred) of stimulus Sets, half of them were the Lead-ball stimulus sets and half of them the Ping-pong-ball stimulus sets, were randomized in order and put into a pile for the experiment. In the experiment, the subjects were first given an information sheet which gave them three key instructions of the experiment:
"1) the experiment is not a test;
2) use your gut-feeling to answer the questions;
3) do not go back to any completed answer."
To ensure all subjects attended to these instructions, the experimenter also pointed out and read out loud these instructions to the subjects. The experimenter then randomly distributed the pile of stimulus sets to the subjects. On every page in each set, the subjects were instructed about the basic information of the experiment: a spiral tube was lying flat on a smooth table and the subject was looking down on the surface of the table as he or she was looking at the graph. Depending on the conditions, the subject would be instructed that either a ball, or a pile of sand, or a gush of water was shot into the tube and was asked to indicate which white ping-pong ball would get hit when it comes out the tube. (For example, under the Lead-ball and Fast condition, the subject was instructed that a lead ball was shot into the tube and it traveled at high speed inside the tube; He or she was then asked to indicate which white ping-pong ball would get hit by the lead ball after it came out the tube. When the subject finished his or her last page in the set, the experiment came to a close. The same procedure was used for every group of subjects, until 134 students were tested on stimulus Set 1 and 129 on stimulus Set 1.
Design: There were two between-subject conditions: the Lead-ball or LB condition and the Ping-pong-ball or PP condition. Stimulus Set 1 contained the stimuli for the LB condition and stimulus Set 2 the stimuli for the PP condition. There were three within-subject conditions under the LB condition : Fast-ball (Fast), Slow-ball (Slow), and Water; there were four within-subject conditions under the PP condition: Fast-ball (Fast), Slow-ball (Slow), Sand, and Water. As a result, there were altogether six conditions in this experiment: Fast/LB, Slow/LB, Fast/PP, Slow/PP, Sand, and Water. The within-subject conditions were presented to the LB group in the order of Fast-Slow-Water and to the PP group in the order of Fast-Slow-Sand-Water.
Subjects' responses were scored according to which of the 17 white ping-pong balls the subjects indicated would be hit by the mass ejected from the tube. Responses were analyzed by condition. The response frequencies are given in Table 1 for all conditions of the experiment. The frequency distribution histograms are shown in Figure5. Several further results can be obtained from these data.
1. Which Group Is More Physically Correct?
The physically correct answer is the white ping-pong ball at the center position, which is true for every conditions in the experiment.
By numbering the white ping-pong balls (or the measurement positions) 1 through 17 from the first one on the left to the last one on the right, the center position is then position number 9. As a response deviates from position 9, it then becomes physically incorrect to different degrees according to how large the deviation is in number of positions away from the center position. The incorrectness of a response is independent of whether it is at a left position (positions 1 through 8) or a right position (positions 10 through 17). For example, a response at position 8 is as incorrect as a response at position 10, because they both have the deviation of 1 position. Similarly, responses at positions 7 and 11, 6 and 12, ... 1 and 17 all have the same degrees of incorrectness, respectively.
Under such a categorization of the data, the responses under each condition can be seen as having a particular distribution in term of physical correctness. Since a higher degree of incorrectness corresponds to a greater amount of curvature in a response, a statistical comparison between the responses under one condition with those under another condition would tell us whether the responses differ between conditions in term of either physical correctness distribution or curvature distribution.
Our main concern was whether the lead-ball responses are overall the same as the ping-pong-ball responses. Chi-square tests were applied to compare the lead-ball responses and the ping-pong-ball responses categorized by the 9 levels of correctness under both the Fast condition and the Slow condition. In the actual tests, responses at the three lowest levels of correctness, i.e., responses at positions 1, 2, 3, and 15, 16, 17, were further collapsed into one single category in order to satisfy the requirement of a Chi-square test (i.e., the number of cells which have frequencies less than 5 can be no more than 20% of the total number of cells). The responses between the Fast/LB condition and the Fast/PP condition were found to differ significantly (X2= 15.66, p=0.016) and the responses between the Slow/LB condition and the Slow/PP condition, on the other hand, have shown no significant difference (X2=4.26, p=0.642). Therefore, under the Fast condition, the ping-pong-ball group and the lead-ball group had responded quite differently by a measure of physical correctness/curvature distribution; in comparison, under the Slow condition, the ping-pong-ball group and the lead-ball group had not responded differently. As the responses under the Fast/PP condition are more centered around position 9 than those under the Fast/LB condition, we can further draw the conclusion that the ping-pong-ball group is overall more physically correct than the lead-ball group under the Fast condition; in comparison, no such a conclusion can be drawn about the responses under the Slow condition.
2. Further Differences Between Lead-ball and Ping-pong-ball Responses
The above results have provided to us only an overall difference between the lead-ball responses and the ping-pong-ball responses. The characterization of responses by physical correctness is unable to reveal to us what the exact differences are between the responses under the LB condition and those under the PP condition. A more detailed analysis is necessary.
Examining Figure 5, we notice three important facts in the responses under the Fast condition: 1) the number of ping-pong-ball responses at the center position is much higher than that of the lead-ball responses (61 vs. 38); 2) more lead-ball responses curve to the left than ping-pong-ball responses (65 vs. 49); 3) the numbers of responses which are curved to the right between the LB condition and the PP condition are quite close (26 vs. 24). Thus, it appears that it is both the different numbers of responses at the center position and the different numbers of responses curving to the left between the lead-ball responses and the ping-pong-ball responses which are responsible for the overall difference found above. More detailed results, therefore, can be obtained if we compare the lead-ball group and the ping-pong-ball group in terms of their different numbers of curved responses and different numbers of straight responses. In addition, as can be also seen, while the numbers of straight responses between the Fast/LB condition and the Slow/PP condition are fairly close (38 vs. 43), the numbers of the left-curved responses (65 vs. 53) and the numbers of the right-curved responses (26 vs. 36) between these two conditions differ much more noticeably. Therefore, the different numbers of left-curved responses and the different numbers of right-curved responses between the Fast condition and the Slow condition may also be important for finding out more detailed results in this experiment.
Because of the large number of measurement positions (17 of them) in the stimuli, responses at positions 8 and 10, being extremely adjacent to the straight response, differ from the straight response only by a very small angle. As a result, to account for any subject variation of representing a straight response, it is quite reasonable that we treat all the three positions at the center, i.e., 8, 9, and 10, as 'straight' positions and to categorize all the responses at these positions as straight. Any response which is not at any one of these positions is then curved. That is, the subjects' responses are characterized as either straight, i.e., being at positions 8, 9, or 10, or curved, i.e., being at positions 1 through 7 and 11 through 17. For a curved response, it can be characterized as either left-curved, i.e., being at positions 1 through 7, or right-curved, i.e., being at positions 11 through 17.
Chi-square tests have been applied to the subjects' responses under the curved-straight categorization to compare the lead-ball group and the ping-pong-ball group. The lead-ball group and the ping-pong-ball group were found to have responded significantly differently under the Fast condition (X2=4.58, p=0.032); On the other hand, no significant difference was found between the lead-ball group and the ping-pong-ball group under the Slow condition (X2=1.54, p=0.214). These results, together with the results by the correctness measurement earlier, are presented in Table 2.
The same Chi-square tests across the Fast and Slow conditions indicate that the lead-ball group under the Fast condition did not respond differently from the ping-pong-ball group under the Slow condition (X2=0.002, p>0.95), neither did the lead-ball group under the Slow condition respond differently from the ping-pong-ball group under the Fast condition (X2=0.71, p>0.3). These results are listed in Table 3.
The other way of finding out the difference between the lead-ball group and the ping-pong-ball group, as mentioned earlier, is to test the difference in the numbers of left-curved responses in comparison to the difference in the numbers of right-curved responses. By eliminating all the responses at the center positions, i.e., 8, 9, and 10, we compared the lead-ball group and the ping-pong-ball group with respect only to the left-curved and right-curved responses. Chi-square tests across the Fast and Slow conditions indicate, as shown in Table 3, that the lead-ball group under the Fast condition responded significantly differently from the ping-pong-ball group under the Slow condition (X2=6.28, p<0.02); But, the lead-ball group under the Slow condition however did not respond differently from the ping-pong-ball group under the Fast condition (X2=0.13, p>0.7).
3. Within-subject Differences
Did a subject respond differently under the different conditions in a stimulus set? Recall, there are three within-subject conditions (i.e., Fast, Slow, and Water) for the lead-ball group and four within-subject conditions (i.e., Fast, Slow, Sand, and Water) for the ping-pong-ball group. Comparing the responses between any two within-subject conditions of a group, we were able to decide whether or not a subject within each group responded differently to different conditions. McNemar tests have been applied to the data categorized in the curved-straight fashion.The results of such comparisons are shown in Figure 6. Table 4 further provides the significance values of these tests. It can be seen that, overall, the subjects in both groups responded differently to different conditions. Except for the Fast-Sand comparison in the ping-pong-ball group and the Fast-Slow comparison in the lead-ball group, all the test results have turned out to be significant.
4. Differences Among Subjects
Significant differences between men and women were noticed in this experiment under the Fast condition and the Water condition. Multiple Linear Regression Analysis shows that the difference is significant for both the Fast condition (p<0.001) and for the Water condition (p<0.05). This indicates that significantly more women have drawn curved responses under both conditions.
Under the Sand condition, no significant difference has been identified between men and women, but a significant difference in the age of the subjects has been found (with p=0.002). Such a result appears to show that a higher age is correlated to a more accurate response under the Sand condition (the age of the subjects in this experiment ranges from 17 to 38 years).
The differences found between the responses under the LB condition and those under the PP condition indicate that the subjects were cognitively sensitive to the dynamic variable of Mass. The difference between the responses under the Fast condition and those under the Slow condition demonstrates that the subjects were cognitively responsive to the kinematic variable of Speed. The effects of Water and Sand found in the subjects' responses testify the existence of cohesiveness representations in the cognitive system of the subjects. Now , let's look at these results in more detail.
Sensitivity to Levels of Mass:
The conditions of PP and LB in this experiment were designed to test the effect of Mass. That the subjects were found to respond significantly differently under the Fast/PP condition from under the Fast/LB condition shows that the subjects have cognitively distinguished a ping-pong ball from a lead ball. Since, in the stimuli, the lead ball and the ping-pong ball were completely identical in term of visual information, they are then cognitively distinguishable only by their difference in Mass, i.e., dynamic information.
The differences found between the responses under the Fast/PP condition and the responses under the Fast/LB condition have shown to us that a lead-ball response is more likely to be curved than a ping-pong-ball response. In addition, as the curved lead-ball responses were found to curve mostly to the left, the subjects must have conceived the lead ball as more likely to curve to the left, i.e., in the curving direction of the tube. These results therefore suggest that the cognitive system conceives a ball of heavy mass as more likely to continue a curved motion and a ball of light mass on the other hand as more likely to dismiss a curved motion and renew into a straight one.
Such an unique distinction between a lead-ball response and a ping-pong-ball response under the Fast condition hence demonstrates not only the existence of representations for the different levels of Mass within the cognitive system and but also the involvement of such representations in the subjects' conception of motion. Since mass, in contrast to visual information, is purely dynamic information, we thus have evidence that the cognitive system had in fact incorporated dynamic information into its conception of spatiotemporal information, i.e., conceiving motion as dynamic.
Such a conclusion further confirms the main finding of our last experiment in which the subjects were also found to have responded to the levels of dynamic information. In the present experiment, however, we will further investigate how exactly the subjects responded to the levels of dynamic information.
Sensitivity to Levels of Mechanical State:
It is quite appealing to try to describe a curved response in term of impetus, because it seems to follow the prediction of the Impetus Theory. However, the fact that it is the variable of Mass which had caused a significant effect in the present experiment urges us to look for a more detailed theoretical explanation. An alternative is to adopt the physical notion of mechanical state.
Physically, the mechanical state of a moving object can be precisely described by the kinematic and the dynamic characteristics of the object. In our present experimental setting, Mass is the dynamic variable and Speed is the kinematic variable. While the PP condition and the LB condition were to test the effect of Mass, the Fast condition and the Slow condition tested the effect of Speed.
Recall, a significant difference was found with respect to Speed in the ping-pong-ball group (see Figure 6), but not in the lead-ball group. In addition, significant differences were found between the lead-ball responses and the ping-pong-ball responses under the Fast condition, but not under the Slow condition (see Table 2). A significant difference was further found between the lead-ball responses and the ping-pong-ball responses across the Fast and Slow conditions under the left/right characterization (see Table 3). Therefore, the experimental results involving the variables of Mass and Speed strongly suggest that the subjects were sensitive simultaneously to both the mass and the speed of the ball. In order to find a more adequate theoretical explanation of these results, the effects of Mass and Speed must be considered together.
Figure 7 illustrates six comparisons, represented by A, B, C, D, E, and F, among the four conditions of the experiment involving the lead ball and the ping-pong ball. Here, each condition, which is a combination of a value of Speed (either high or low) and a value of Mass (either heavy or light), can be regarded as corresponding to an unique mechanical scale or state instantiated in the current experimental stimuli. Namely, Slow/PP has the lowest the mechanical scale in comparison to any other Mass/Speed combinations in Figure 7. Moving up on the mechanical scale along the dimension of Speed, we have the mechanical state of Fast/PP. Likewise, the mechanical state of Slow/LB is one level higher than the mechanical state of Slow/PP along the dimension of Mass. Fast/LB has the highest mechanical state, since it is one level higher than the mechanical state of either Slow/LB or Fast/PP and two levels higher than the mechanical state of Slow/PP.
The significance/non-significance of the curve-straight tests involving the lead-ball and the ping-pong-ball responses (from Tables 2, 3, and 4) are shown in the figure. Two important points can be made with respect to Figure 7.
First, the subjects were cognitively sensitive to the mass as well as the speed, i.e., to the mechanical state, of the ball. As can be seen in the figure, the effect of Mass is significant only when Speed is high and the effect of Speed is significant only when Mass is light. Heavy Mass seems to have a greater influence on the subjects' conception of a strong mechanical state than high Speed and low Speed seems to play a greater part in the subjects' conception of a weak mechanical state than light Mass, as shown by the significance of tests B and D in contrast to the non-significance of tests A and C. In other words, in conceiving the mechanical state of Fast/LB, the subjects more saliently represented the dynamic content of mass; while in the conceiving the mechanical state of Slow/PP, the subjects were more sensitive to the spatiotemporal information of speed.
Second, as can be seen in the figure, the significant difference between the mechanical states of Fast/PP and Fast/LB and the significant difference between the mechanical states of Slow/PP and Fast/PP suggest that the subjects must have treated the mechanical state of Slow/PP as different from the mechanical state of Fast/LB. However, as shown in the figure, the comparison (E) between the conditions Fast/LB and Slow/PP has not produced any significant difference, indicating that the subjects had not treated the mechanical states of Slow/PP and Fast/LB as different. The resolution to such a conflict can be found in the significant difference between the responses under the Fast/LB condition and the responses under the Slow/PP condition in Table 3, which indicates more responses under the Fast/LB condition are curved to the left while more responses under the Slow/PP condition are curved to the right. (Such a difference can be seen quite directly in Table 1, where more responses curve to the right under the Slow/PP condition than under the Fast/LB condition and more responses curve to the left under the Fast/LB condition than under the Slow/PP condition, while the numbers of straight responses (i.e., at positions 8, 9, and 10) remain about the same (69 vs. 71) under both conditions.) Therefore, between the two extreme mechanical states of Fast/LB and Slow/PP, two opposite patterns had occurred concerning the number of left-curved responses and the number of right-curved responses. In other words, although both were found to be different from a response under the Fast/PP condition, a response under the Fast/LB condition and a response under the Slow/PP condition were in fact different from it for exactly the opposite reasons. This fact strongly suggests that the subjects had treated the mechanical state of Fast/PP as in-between the mechanical state of Fast/LB and the mechanical state of Slow/PP and had reacted to the stimuli according to the levels of the mechanical state thereof.
With reference to Experiment 1, the subjects in this experiment were therefore found to be cognitively sensitive to the mechanical level of an object in motion. The dynamic level of the last experiment has now been extended to the mechanical level in this experiment which are characterized by both the dynamic variable of Mass and the kinematic variable of Speed.
The Representation of Mechanical Magnitude:
Although the cognitive system was found to be sensitive to Mass and Speed, the mass and the speed of an object have to be represented by the cognitive system with reference to the single object in motion. That is, the representation of mass and the representation of speed are in fact intimately related to each other and bonded in a single representation of dynamic motion. In order to capture the integrated representation between mass and speed, we need to adopt another physical concept --- mechanical magnitude.
Physically, the mechanical magnitude of motion is the magnitude of momentum, mv, which is characterized by its linear relationships to the mass m and the speed v of an object in motion. Cognitively, as the subjects were found to be more sensitive to Mass than Speed under the Fast/LB condition and more sensitive to Speed than Mass under the Slow/PP condition, the subjects' representation of the magnitude of dynamic motion obviously does not relate linearly to their representation of mass or their representation of speed. In other words, although both are related to the mass and the speed of an object, the subjects' conception of the object's mechanical magnitude is related to them quite differently from the mechanical magnitude itself. Nonetheless, an unique conceptual integration between the representation of mass and the representation of speed appear to render the cognitive system a representation for the strength of dynamic motion when it conceives an object of high mechanical magnitude as more continuous or an object of low mechanical magnitude as more deviant.
In the following, I will use the term the representation of mechanical magnitude to stand for the magnitude or strength of dynamic motion as represented by the cognitive system.
The Representation of Continuity Constraint on Dynamic Motion:
According to what we have found, a ball of heavy Mass at high Speed is correlated to a significantly higher proportion of left-curved responses, while a ball of light Mass at low Speed is correlated to a significantly higher proportion of right-curved responses. Since Fast/LB and Slow/PP, respectively, correspond to the highest and the lowest mechanical levels of the mechanical state of the stimuli, curving to the left and curving to the right are therefore the two opposite effects with respect to the mechanical levels of the stimuli. We have further found that a representation of heavy mass contributes more than a representation of high speed to the representation of mechanical magnitude of a heavy object at high speed; a representation of low speed contributes more than a representation of light mass to the representation of mechanical magnitude of a slow object with light mass. Finally, the subjects seem to have treated the mechanical state of Fast/PP as in-between the mechanical states of Fast/LB and Slow/PP.
The higher number of left-curved responses under the Fast/LB condition indicates that the subjects conceived a heavy ball at high speed as more likely to continue its own curved motion. The higher number of right-curved responses under the Slow/PP condition shows that the subjects conceived a slow ball with a light mass as the most likely to deviate from its own past motion. Interestingly, the subjects conceived the motion of a ball as the most continuous under the condition of Fast/LB where they were found to be more sensitive to Mass than Speed; They conceived the motion of a ball as the least continuous, i.e., the most likely to deviate from its own past motion, under the condition Slow/PP where they were found to be more sensitive to Speed than Mass. These results seem to suggest that it is by cognitively incorporating more mass or dynamic information that the cognitive system conceives the dynamic motion of an object as more continuous; and, it is with the pure conception of the speed or spatiotemporal information that is the cognitive system likely to represent a motion path as deviant. The following analysis should make this point more clear.
As the subjects were found to conceive the motion of a ball under the Fast/PP condition as in-between the motion under Fast/LB condition and the motion under Slow/PP condition, i.e., as traveling straight, we can clearly see an interesting pattern in the responses between these three conditions. Since the difference between the mechanical states of Fast/PP and Fast/LB lies exclusively in the different levels of Mass they each have and the difference of the responses under the Fast/LB condition from those under the Fast/PP condition is essentially their left-curvedness, the representations of the left-curved responses under the Fast/LB condition have to be strongly related to the subjects' representation of the heavy Mass of the lead ball in this condition. In contrast, since the difference between the mechanical states of Fast/PP and Slow/PP lies exclusively in the different levels of Speed they each have and the difference of the responses under the Slow/PP condition from those under the Fast/PP condition is characteristically their right-curvedness, the representations of the right-curved responses under the Slow/PP condition have to be strongly related to the subjects' representation of the low Speed of the ping-pong ball in this condition.
It appears, therefore, that the cognitive constraint for the continuity of motion is strongly connected to the subjects' representation of the dynamic information of mass; The representation of the ball's deviation from its own past motion, which attests a freedom from the cognitive constraint for the continuity of motion, corresponds to the lack of a representation of mass and therefore can be uniquely traced to the representation of the spatiotemporal information.
But, how does the representation of mass participate in the conception of motion in the cognitive system? Following the above discussion, a representation of heavy mass appears to be directly translated by the cognitive system into a representation of a mechanical tendency of the object to continue its own way of motion; no such translation seems to exist for a representation of light mass. Therefore, such a cognitive translation appears to operate between the representation of mechanical magnitude and the representation for the continuity of motion.
Recall, the representation of strong mechanical magnitude is an integration between the representation of mass and the representation of speed. The representation of the continuity of motion requires the cognitive system to take into account an object's previous motion trajectory and predict its future motion trajectory. The cognitive system then seems to regulate the degree of continuity according to the representation of mechanical magnitude. In other words, it predicts the future trajectory of a motion from the past trajectory and the strength of the motion --- the stronger the mechanical magnitude is, the more closely the future trajectory resembles the previous trajectory, i.e., a continuation of it.
It is with such an intricate representational complex, which incorporates the information of the object's mass, speed, and past and future motion states, which seems to be the cognitive mechanism that governs the responses of the subjects in this experiment. Conceptually imposing a continuity constraint on the representation of the trajectory of motion and regulating the amount of continuity according to the representation of mechanical magnitude is the unique function of such a representational complex. In the following, I will refer to it as the representation of continuity constraint on dynamic motion, or simply, as the Continuity Representation.
Finally, recall, in Experiment 1, we have come to the conclusion that the higher the dynamic level of the stimulus, the more constrained to are the responses of the subjects; On the other hand, the lower the dynamic level of the stimulus, the larger variations or the less mechanically constrained are the responses of the subjects. In this experiment, not only are the levels of the mechanical state specifically characterized by the values of Mass and the values of Speed, the mechanical constraints and variations of the responses are also concretely measured by the responses of the subjects on the measurement positions. With more accurate measurements, we have been able to furnish a much more detailed theoretical explanation about the representation of dynamic motion.
A Dynamic Explanation of Sand and Water Responses:.
Sand is non-cohesive and non-rigid, i.e., it is composed of extremely small particles that do not stick together. The answer to the question concerning why the subjects had responded to a ping-pong ball and sand similarly, I believe, lies in if the subjects really conceived of sand as such, i.e., as containing small particles of little mass, which non-cohesively join together in space and time. Such a conception of sand would make the handful of fast moving sand lower on the mechanical scale than a fast moving lead ball, i.e., similar to a fast moving ping-pong ball. Recall, the ping-pong ball was conceived of as incapable of assuming a high mechanical state mainly because it lacks mass, as a result, the ping-pong-ball responses were found to be quite different from the lead-ball responses. I now argue that the non-cohesive sand, rather, its small and indefinite particles were also not conceived of as having enough mass to have the same mechanical level as a lead ball.
Water can also be characterized in the same way as sand. In Table 1, the responses under the Water condition can be seen to have the highest level of correctness. In Figure 6, water responses are shown to differ significantly from all other types of responses due to such a high level of correctness. This result that the subjects were found to respond to the water stimulus most likely with a straight drawing turns out to be consistent to the results found in (Kaiser, Jonides, and Alexander, 1986).