Newton’s cradle and Astro-Blaster: toy hybridization for studying chain collisions

We present a multiple-collision apparatus, based on low-cost and easy-to-find materials, which offers the opportunity to plan and conduct a sequence of experimental activities aimed at studying, in a very simple way, the energy transfer in multiple collisions. Our apparatus is based on the use of everyday objects such as toys, which stimulate students’ interest by creating a familiar learning context. Two significant experiments on multiple collisions are described and the analyses of the corresponding experimental results are discussed. The first experiment makes it possible to analyze the kinetic energy transfer along a chain of adjacent balls. The second experiment offers the opportunity to give an analogical conceptual illustration of the bounce of a supernova core after gravitational collapse. Each experiment is pedagogically introduced by asking students to answer an appropriate stimulus questions, which were identified by instructors in a preliminary brainstorm session with students. Last, a theoretical model is given for kinetic energy transfer along a chain of balls and the experimental results are compared with the model predictions.


Introduction
Momentum and energy transfer in collisions represent fundamental learning topics of basic mechanics [1,2]. Moreover, a typical chain of one-dimensional many-body collisions [3] can constitute a useful conceptual metaphor [4] for didactic modeling of different phenomena, ranging from mechanics [5] to optics and acoustics [6], and from electromagnetism to quantum mechanics [7]. For example, the case of recurrent collisions among elastic spheres represents an interesting model for a wide and heterogeneous set of physical systems characterized by the concept of 'impedance' [8][9][10][11], i.e. a pervasive physics concept that controls the matching between subparts of a system and determines the transfer of some physical quantity between interacting subparts.
A unidimensional chain of collisions among elastic balls represents an interesting educational system to promote students' reflection on the theme of coupling between subparts of a system through a succession of elementary interactions. The simplest example of such a chain is represented by Newton's cradle, in which a series of identical metal balls are suspended in a line, just touching each other [12,13]. In this paper, we present some experiments on multiple collisions based on a simple apparatus that can be constructed by using low-cost and easy-to-find materials. Our choice is based on the well-known educational value of using everyday objects in physics teaching/learning processes ( [14] and references therein). Among such objects, toys play a special role, since they make it possible to stimulate students' interest and create a familiar learning context [15][16][17].
From this perspective, we developed a multiple-collision apparatus, based on Newton's cradle, consisting of appropriate sequences of colliding balls obtained from the well-known Astro-Blaster toy [18,19]. The main goal of this paper is to propose two experimental setups that teachers can integrate into learning paths as a part of more general educational activities.
We used the multiple-collision apparatus to plan a coherent sequence of laboratory activities based on a video-analysis data acquisition system, which offers the opportunity to simplify the analysis of collisions and study their effects on the entire chain in a very simple way. The level of accuracy of this experimental device strictly depends on the characteristics of the videos (in particular on the spatial resolution and on the frame rate). In our case, we used a commercial-grade high-speed camera to set up quantitative experiments on the transfer of energy in multiple collisions, fully exploring the dependence of the transfer efficiency on the ratio of involved masses. It is important to specify that many modern smartphones offer the opportunity to record significant videos (in terms of resolutions and frame rate) and, consequently, to perform quantitative experiments with a significant level of accuracy.
The paper is arranged as follows. In section 2, the experimental apparatus is described. In section 3, two significant experiments are outlined and the analyses of data are discussed; in this section, reference is also made to the involvement of a group of high school students in the pedagogical tuning of the experimental configurations and procedures. In section 4, the conclusions are deduced and the possible educational implications are illustrated. Moreover, two preliminary models and experiments (which represent the 'conceptual building blocks' for the two setups discussed) are provided in appendix A, in order to prepare students for the complete analytical model provided in appendix B (which explains the empirical results obtained in one of the proposed experiments).

Experimental apparatus
We arranged the multiple-collision apparatus by 'merging' two well-known educational toys: a Newton's cradle [12,13] and an Astro-Blaster TM [18,19], which 3 is also known as 'Seismic accelerator' 4 . The Astro-Blaster used in this work (described in [19]) is composed of a kit including four superballs (figure 1), whose characteristic parameters are summarized in table 1, together with those of two additional superballs purchased in a local toy store.
Among the superball properties given in table 1, a relevant parameter is the coefficient of restitution (COR), which is the ratio of the final to initial relative speed between two objects after their collision. Following the procedure suggested by Bernstein [20], CORs were measured by dropping each superball on a rigid marble surface.
The toc-toc sound sequence of multiple bounces of a given ball (recorded by means of a smartphone audio recording app) was analyzed by means of the free audio analysis software Figure 1. The Astro-Blaster toy, also known as Seismic Accelerator. The toy kit provides four superballs, whose characteristics are summarized in table 1. Audacity 5 , in order to obtain the appropriate plot (such as that given in figure 4 of [20]), from which we obtained the corresponding COR. The experimental apparatus is essentially Newton's cradle (figure 2) in which the ball sequence can be modified as needed. The balls are bifilarly suspended (by means of thin inextensible wires) to two parallel threaded rods, on which the wires are fixed by pairs of bolts. By adjusting the bolts, the position of the balls can be finely tuned in order to align their The modified Newton's cradle. The balls are bifilarly suspended by means of thin wires (inset b: side view) to two threaded rods, on which the wires are fixed by pairs of bolts (inset a). By adjusting the bolts, the position of the balls can be finely tuned in order to align their centers and to make them almost in contact when hanging at equilibrium. The cradle can be implemented in many ways by using different sequences of balls. Two specific sequences (S) were suggested to the students in our work. centers and to make them almost in contact when hanging at equilibrium. The cradle can be implemented in many variants, by using different sequences of balls, in order to investigate different aspects of multiple collisions. In our trials with a group (N = 25) of fourth-year high school students, we used the two sequences shown in figure 3: -Sequence 1 (S1), aimed at investigating the symmetry properties of the kinetic energy transfer between the first and last ball of a chain (experiment described in paragraph 3.1); -Sequence 2 (S2), designed to give an analogical conceptual illustration of the bounce of a supernova core after gravitational collapse [21,22] (experiment described in paragraph 3.2). S1: two balls (GO and MC, see  Sequences S1 and S2 were used to design two specific experiments that were suggested to students in order to answer some sample questions that arose from a brainstorming session on the modified Newton's cradle (see next: figures 4 and 6). The experiments are described in section 3; the corresponding theoretical models are outlined in appendix B.

Two experiments on multiple collisions
Multiple collisions among many balls represent very complex phenomena for students. Some preliminary simpler experiments could help them to understand these kinds of phenomena. From this perspective, we propose in appendix A, two preliminary models and experiments which represent the 'conceptual building blocks' for the two setups discussed, respectively, in sections 3.1 and 3.2.
The modified Newton's cradle was presented to students during a brainstorming session, which allowed identifying two questions, subsequently used as stimulus questions (SQs) for introducing the experiments inspired by them. This experiment was introduced by asking students to answer the SQ illustrated in figure 4, with the following results: 18 students out of 25 (72%) answered D, five students (20%) answered A and two students (8%) answered C. In other words, the majority of students had no idea how the energy transfer through collisions works; some students intuitively answered that heavier balls are more efficient in transferring their energy to lighter ones, than vice versa.
To experimentally investigate SQ1, two runs were performed with the modified Newton's cradle, equipped with the two ball sequences given in figures 4(a) and (b), respectively. Each run was filmed at high-speed using the same equipment and following the procedure described elsewhere [23], except for the video camera model. In fact, to attain a higher frame rate, in the present work a commercial Nikon 1-V3 digital camera was employed, capable of a frame rate of up to 1200 fps. The video analysis of the movies was done following [23] and the results are plotted in figure 5.
Linearly fitting the portions of plots in figure 5 corresponding to uniform motion, the following values (table 2) are obtained for the speed and kinetic energy of the in-and out-ball in the two cases: It is worth observing that we consider the intermediate sequence of balls (from ball 1 to ball n-1) as a whole, acting as a 'mechanical coupler' between two extra balls (i.e. ball zero or 'in-ball' and ball n or 'out-ball'). From this perspective, we focus the analysis on the two extra balls, neglecting the mechanical state of the transmission chain after the collision (even if the chain has a certain amount of momentum and of both kinetic and thermal energy).
The experimental results in table 2 give the correct answer to SQ1: Indeed, this result (counterintuitive, at least for our students, considering their answers to SQ1) can be obtained through a simple analytical model, given for completeness in appendix B. Moreover, the experiment on kinetic energy transmission allows for estimating the speed of propagation of the shock along the ball chain, a topic addressed in theoretical studies [24]. In fact, the time interval taken by the shock to propagate through the chain can be estimated as the delay between the stopping of the in-ball and the start of the out-ball (τ values in figure 5), obtaining: t » 11 ms 1 and t » 9 ms. 2 Considering the length of the ball chain and taking as propagation time the mean value between t 1 and t , 1 the following estimate is obtained for the shockwave speed along the chain: » v 24 m s . s 1 This result can be usefully compared to the speed of sound in rubber, that is of the order of 40-60 m s −1 : the order of magnitude is the same, but the actual value of the shockwave speed is smaller than the typical speed of sound.  Table 2. The speed of the in-and the out-ball in the two cases given in figure 4, obtained by fitting the video-analysis data plotted in figure 5. The corresponding value of the kinetic energy allows obtaining (last column) the fraction of kinetic energy of the hitting (in) ball, that is transferred to the last (out) ball. Fractions are equal within experimental errors, which are of the order of 5%.
This can be interpreted in terms of some delay in the propagation of the shock along the chain, due to the interaction between adjacent balls with respect to a continuous piece of rubber having the same length.
3.2. Experiment 2: a toy model for supernova core bounce.
This experiment was introduced by asking students to answer the SQ illustrated in figure 6, with the following results: 12 students out of 25 (48%) answered A, eight students out of 25 (32%) answered B, three students (12%) answered C, two students (8%) answered D.
Comparing with the predictions of Experiment 1, in this case students were globally more confident in guessing a result. In fact, only two students did not make a prediction at all. Interestingly, it seems that students identified the mass as the main characteristic of a ball that makes it capable of 'carrying' kinetic energy, as suggested by 32% of students answering B versus only 12% answering C. The discussion with students of the results of SQ2 highlighted the need for a better definition of the concept of 'effectiveness', of a given kind of ball, in taking away the kinetic energy of the system. After some debate, the attention was focused on the opportunity to relate the energy transported by a certain ball to its mass, i.e. the energy transported per unit of mass. The following definition was then proposed for the 'effectiveness', η: h = FRACTION of the incoming energy, taken away by a given mass FRACTION this mass represents with respect to the total mass . Analytically (taking the green (G) ball as an example): is the kinetic energy taken away by the single ball and K in is the total kinetic energy of the incoming balls (the mass symbols are defined in table 1).
To experimentally investigate SQ2, the modified Newton's cradle was implemented with the ball sequence S1 shown in figure 3(b). Such arrangement allows exploring the symmetric collision of two Astro-Blasters ( figure 7). However, from a strictly experimental point of view, it is hard to implement a perfectly symmetric initial condition-indispensable to obtain an appropriate central collision. This problem can be overcome by noting the mechanical equivalence between: (i) the symmetric knock-out of two opposed Astro-Blasters and (ii) the collision of a single Astro-Blaster against an infinite wall (i.e. a wall whose mass is much larger than the Astro-Blaster mass). To be precise, this equivalence holds if the COR for the ball-ball collision and the COR for the ball-wall collision are equal. Indeed, this is our case, within few percent, since all estimated CORs (between balls and between the blue ball and the wall-see figure 8) are greater than 0.85.
The whole sequence lasts about 15 ms. The time step between adjacent frames is not constant.
The video analysis of the movie (figure 8 shows some frames) allows determining the speed values of the balls (before and after the impact) according to the data analysis procedure  Plotting the η values versus the reciprocal mass of the balls, an approximately linear relation can be observed ( figure 9). This suggests that external masses of the Astro-Blaster balls are more efficient in taking away kinetic energy from the system, roughly on an inverse proportionality base. The data in table 3 indicate, for example, that the external, lightest, red ball has a (mass-specific) efficiency in taking away kinetic energy that is about 19 times (=7/ 0.37) that of the heaviest internal blue ball. In other words, 22% of the total kinetic energy is taken away by 59% of the colliding mass (B), where 28% of kinetic energy is taken away by only 4% of the mass (R).
The Astro-Blaster toy was suggested [18,21] as a useful mechanical analogy to conceptually illustrate the mechanism governing the catastrophic expulsion of the outer shell of a supernova during the stellar explosion. In particular, Egler [21] proposed a demonstration based on the constitutive unit of an Astro-Blaster, i.e. a couple of elastic balls having different masses (a softball and a basketball). Some years later, Mancuso and Long described the  Astro-Blaster [18], reporting a quotation by the distinguished astrophysicist Dr Stirling Colgate 6 , who is credited as the scientific creator of this toy: 'Astro-Blaster illustrates the laws of conservation of momentum and energy during the creation of a supernova (an old star, that having exhausted its nuclear fuel, collapses upon itself in less than a second). A shockwave speeds outward from the center through the collapsed material, moving faster and faster it reaches less dense layers toward the surface. The shockwave accelerates an outermost thin layer of the collapsed star to relativistic speeds, creating 'cosmic rays' that spread throughout our Galaxy. The Gravitational collapse of the dying star is illustrated by Astro-Blaster's fall to the surface. The shockwave accelerating outward through the star is illustrated by a wave of increasing speed as the result of the impact which is felt by the lighter balls near the top. The supernova explosion and release of cosmic rays is illustrated by the rapid departure of the top ball at high speed.' In order to make students understand the reason for the name of the Astro-Blaster game, they were asked to read the beautiful article published in Scientific American by Bethe and Brown [22], in which the mechanism of the 'rebound' (of the external layerswith lower densityof a star) is described in a simple and rigorous way.
Even though theoretical quantitative studies have been already published on the Astro-Blaster [19,25], as we know, our work represents the first example of an Astro-Blaster being experimentally studied from a pedagogical perspective, with particular reference to its use as an analogy for the supernova core bounce.
It is worth observing that our idea of a horizontal implementation of the Astro-Blaster as a modified Newton's cradle (an idea partly put forward by MacInnes [26]) is particularly valuable for experimental quantitative purposes, since it allows neglecting the gravitational potential energy variation of the balls. In fact, the few centimeters of horizontal displacement of the pendulums (during which the motion is analyzed) allows neglecting vertical displacement, provided that the suspension ropes are sufficiently long (as in our case, where the vertical projection of a suspension rope is about 50 cm).

Conclusions
We present a home-made apparatus based on the merging of two well-known scientific toys, and an experimental methodology based on high-speed video analysis, enabling significant learning activities in order to explore sequences of multiple collisions among elastic balls. The use of a commercial-grade high-speed camera allows performing quantitative experiments aimed at focusing students' attention on the role of transmission chains in transferring physical quantities (particularly, kinetic energy in the present case) from an initial to a final state. Proposed experiments (whose results are partially compared with the predictions of an appropriate theoretical model) highlight the role of the mass ratio sequence in determining the transfer efficiency, by stressing in particular, the symmetry properties of the involved transfer function. By changing the sequence of colliding balls, the proposed apparatus can be implemented in different ways, in order to quantitatively explore various aspects of chain collisions treated in the literature [e.g. 3,6,8,10,18,19,21,24,25,27]. Among the many possible sequences, this paper deals with two particularly significant sequences, identified by the instructors following a brainstorming session proposed to the students in relation to multiple collisions. We hope this paper could be inspiring for teachers (both in upper high school and in introductory university courses), since we show how an appropriate implementation of simple toys can be used to experimentally investigate physical phenomena apparently not affordable in a didactic lab, such as the mechanical aspects of a supernova explosion. Our activities can be integrated into different learning paths designed by teachers, considering the specific needs of their learners. From this perspective, the simplified models and experiments proposed in appendix A could help teachers gradually introduce learners to the modeling of the multi-collision systems.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix A. Two simple models
A direct approach to the model for multiple collisions proposed in this paper could be slightly difficult for high school students. The preliminary analysis of a simpler case consisting of only two colliding spheres could be an appropriate didactic strategy to introduce the model and in particular to introduce the concept of 'transfer function' as intended here. In this appendix, we propose, for each of the experimental setups described in sections 3.1 and 3.2, the corresponding 'conceptual building block' consisting of the theoretical model for the simplest case, together with the corresponding experiment.

Experiment 1: a moving ball hits a ball at rest
The first experiment regards the unidimensional anelastic collision of a moving ball against a ball initially at rest (figure A1). This introductory activity represents the building block for 'Experiment 1: kinetic energy transmission' described in section 3.1.
The equations for the collision (respectively, the conservation of momentum and the anelasticity expressed through the COR) are: where COR e is the ratio between the modulus of the relative velocities (after and before the collision). Considering the collision as a dynamic process leading to the transfer of some kinetic energy from the ingoing ball (ball 1 before the collision) to the outgoing ball (ball 2 after the collision), the goal is to obtain an algebraic relationship between the kinetic energy of the outgoing ball and the kinetic energy of the ingoing ball. Since in a unidimensional collision  ¢ ¢ v v 1 2 (i.e. the impacting ball either moves backwards or keeps moving forward with a slower speed than the impacted ball, depending on the mass ratio) equation (A1) can be easily solved as: 2 ) can be expressed as: where T K is a function of the mass ratio ( / = r m m 1 2 ) and of the COR e: This function expresses the fraction of the kinetic energy of the incoming ball that is transferred to the outgoing ball in the collision: in order to highlight its physical meaning, we can name T K as the energy transfer function for the collision. Figure A2 shows the dependence of the transfer function on both the mass ratio and the COR. As expected, the kinetic energy transfer increases with increasing values of e (i.e. as the collision becomes more and more elastic). An interesting feature is represented by the maximal efficiency in the energy transfer, which is attained for = r 1 (i.e. equal masses), independently of the COR value. The existence of this maximum follows from a notably symmetry property of T : K Figure A2. Dependence of the energy transfer function on the mass ratio r and on the COR e. The right inset shows a detail corresponding to small r values. The vertical axis is normalized in order to factor out the dependence on e, which contributes an overall scale factor to the transfer function. This property (that can be easily obtained from the expression equation (A4)) has a clear physical meaning: the fraction of transferred energy is invariant under the exchange of the two balls. In fact, exchanging balls corresponds to the following transformation:

A 6
The suitability of the model has been tested by means of the collision experiment between only two balls (illustrated in figure A3 and analyzed through the video-analysis process-see figure A3). Table A1 shows: (a) the experimental parameters (masses and their ratio); (b) the relevant values of the speed obtained by fitting the appropriate motion data (as described in figure A3); (c) in the last two columns, the experimental value for / ¢ K K 2 1 and the value predicted by the model through equation (A4). The agreement between the two data (0,79 and 0,81) is very interesting, considering the experimental uncertainty (the percentage uncertainty for the speed values obtained through the video analysis can be estimated as around 3%).  (table A1). Table A1. Comparison between the experimental value of / ¢ K K 2 1 (obtained by the video-analysis process) and the value predicted by the model (formula A.4), using the COR value = e 0.93 determined from the impact data plotted in figure A3. The second experiment represents the building block for 'Experiment 2: a toy model for supernova core bounce' described in section 3.2. It regards the impact of two (independently) co-moving balls against an 'infinite wall', i.e. a wall whose mass is much greater than that of each ball. The upper portion of figure A4 shows the model and the definition of the used symbols; the lower portion of the same figure shows some corresponding frames from the high-speed video of the simplified experiment. The collision of the balls against the infinite wall has been modeled as a very rapid succession of two partial collisions: (A) the collision leading from the initial state (both balls move towards the wall with the same velocity) to the intermediate state (the ball m β has reversed its velocity, while m α keeps moving forward); and the collision (B), leading from the intermediate state to the final state (both balls move backward).
Assuming the same value for the COR, = e 0.93, for both kinds of impact (green versus blue ball and blue ball against the wall) 7 and considering the conservation of momentum, the equations for the two partial collisions are: . A7 We can assume < r 1 for the purposes of this model (i.e. as a building block for modeling the Astro-Blaster toy in section 3.2) so that the α ball is the lighter and the β ball is`the heavier. Figure A5 shows the dependence of the final velocities on the mass ratio, both for the elastic case (solid lines) and the actual = e 0.93 case (dashed lines). In particular, the graph shows that: (i) the lighter ball moves (always) backward after the impact of the two balls against the wall, while the heavier one moves backward or forward depending on whether * < r r or * > r r , where ( ) * / / = + + @ r e e e 1 1 3; 2 (ii) for equal masses, the two balls have nearly the same final speed (exactly the same in the elastic case), but in opposite directions; (iii) when the mass ratio tends to zero, the speed of the lighter ball tends to be ( ) + e 2 times the heavier one.
Another interesting result of this preliminary problem is the expression for the 'effectiveness' parameters defined in section 3.2 (equation (1)), i.e. the ratios:      Figure A6 shows the dependence of the effective parameters on the balls' mass ratio. The lighter ball is always the most effective (in relation to its mass fraction) in taking away the kinetic energy after the collision; except for equal masses ( = r 1), when the two balls have nearly the same effectiveness (exactly the same in the elastic case). Moreover, as r decreases, the effectiveness of the lighter ball steadily increases with respect to the heavier one. Interestingly, when the mass of the greater ball is just triple the smaller one ( / = r 1 3), h = b 0. Obviously, this is due to the fact that following the impact the heavier ball stops (figure A5). It is worth noting that the true value for which h b vanishes is not just 1/3, but the value * / @ r 1 3 above introduced: this information is lost in equation (A9b) due to the approximation process performed. We tested this model by comparing the predictions of equation (A8) with the experimental results of the impact shown in the bottom portion of figure A4. The balls used for such an experiment are the green and the blue balls listed in table 1 of the main text. The video analysis of the impact (figure A7) leads to the velocity ratios given in table A2.
For the lighter ball (i.e. the Green ball, not directly impacting on the wall) the agreement between the predicted and the observed values is satisfactory (considering 3% uncertainty for the speed values obtained through the video analysis). More complex is the comparison between the experimental and the theoretical values for the blue ball (directly impacting the wall). In fact, the analysis of the B portion of the data in figure A7 gives a negative value for  b v (i.e. the blue ball moves to the left), while the theoretical model predicts the blue ball moving to the right after the first impact with the green ball (i.e. blue ball moves again v v , 0 (obtained by the kinematic data in figure A7) and the values predicted by the model (formulae A.8). The mass ratio for this experiment is = r 0.45 (green and blue balls in table 1 of the main text). ( * ) The experimental value for  b v is not directly measurable, due to the possibility of multiple collisions of the blue ball (back-and-forth) against the green one and the wall (see discussion in the appendix text).

Experimental Predicted
towards the wall). This could suggest that the blue ball undergoes a very fast sequence of multiple collisions, back-and-forth against the wall and the green ball. The possibility to quantitatively resolve in time this sequence of multiple collisions is beyond the technical capability of our camera. However, our video 8 shows clearly at least one back-and-forth movement of the blue ball, though the insufficient time resolution does not allow the quantitative evaluation of the involved speeds. For the sake of completeness, we observe that such a behavior could be ascribed to a mechanism similar to the one highlighted by Cross in 2021 [27]. Indeed, it could be that the green ball initially lags slightly behind the blue ball when they are swinging down towards the wall so that the blue ball definitely collides with the wall and bounces back before it collides with the green ball. Anyway, this possible very fine effect is beyond the spatial and time resolution of our experimental analysis; moreover, our experimental device features insulating suspension wires, which do not allow Cross's electrical monitoring of the contact between balls.

Appendix B
The transfer function for a n-collisions chain The unidimensional chain collision among n + 1 balls (figure B1) can be considered as a process in which a fraction of the kinetic energy of the (only) ball initially moving (m 0 ) is transferred to the last ball (m n ), through the 'transmission line' constituted by the sequence of balls ¼ -m m . .