This paper addresses a popular topic in science teaching and competitions for primary and secondary school students. Experiments with colliding coins are relatively easy to perform and therefore popular in science lessons. We used the idea in the science competition we organized for pupils aged 6 to 13 years. The science competition is based on a few experiments that the participants have to carry out before the competition. Some years ago, 10- and 11-year-old pupils had to experiment with coins. These were 1-, 10-, 20- and 50-euro cent coins (see Fig. 1), but the experiments can also be carried out with quarters and dimes. The experiments were simple elastic collisions with some obvious but also some profound outcomes, which will be described below. Even if the bodies are moving fast and the collision itself is an instantaneous event, it is possible to reconstruct the trajectories from known initial and final positions of the coins. At the end of this paper, we will present the set of problems that were given in the competition.

This paper addresses a popular topic in science teaching and competitions for primary and secondary school students. Experiments with colliding coins are relatively easy to perform and therefore popular in science lessons.^{1–6} We used the idea in the science competition we organized for pupils aged 6 to 13 years.^{7} The science competition is based on a few experiments that the participants have to carry out before the competition.^{8–11} Some years ago, 10- and 11-year-old pupils had to experiment with coins. These were 1-, 10-, 20- and 50-euro cent coins (see Fig. 1), but the experiments can also be carried out with quarters and dimes. The experiments were simple elastic collisions with some obvious but also some profound outcomes, which will be described below. Even if the bodies are moving fast and the collision itself is an instantaneous event, it is possible to reconstruct the trajectories from known initial and final positions of the coins. At the end of this paper, we will present the set of problems that were given in the competition.

## The experiment

The only equipment needed is a set of coins, a large piece of relatively stiff paper (A2 or B2 format), a ruler, and a pencil. First, two parallel lines are drawn on the paper at a distance of about 3 cm, followed by a line perpendicular to these two lines. Two coins are placed on the paper at their initial positions as shown in Fig. 2(a).

The first coin is a projectile, and the other is a target. By flicking the projectile, as shown in Fig. 2(b), the impulse is given to the coin, which starts to move towards the stationary target coin. The coins collide, and after the collision one or both coins move and soon come to rest due to friction on the paper surface. The coins are moving fast and the collision itself is an instantaneous event. The movement of the coins during the collision cannot be observed with the naked eye, but the stationary initial and final position of the coins tell us a lot about the event. Hence, with these experiments, we pay attention to initial and final positions of the colliding coins.

The first task was to practice and learn about central collisions. The perpendicular line along which the projectile and the target were initially aligned served as a tool to distinguish between central and non-central collisions. In the case of successful central collisions, the coins are aligned along this line even after the collision. The central collision of a moving projectile with mass *m*_{p} and a stationary target with mass *m*_{t} has only three qualitatively different outcomes depending on the mass ratio *m*_{p}/*m*_{t} —the mass ratio can be less than 1, equal to 1, or greater than 1. Students are instructed to perform all three variants of central collisions with qualitatively different outcomes, depending on the value of the mass ratio.

A 10-cent coin is taken as a projectile and is flicked first at another 10-cent coin, then at a 50-cent coin, and finally at a 1-cent coin. In the central collision with a coin of the same nominal value (mass), the momentum is transferred from the projectile to the target, and after the collision the projectile stops immediately at the position of collision, while the target moves in the same direction as the projectile was moving before the collision and stops at some distance ahead. If a 10-cent coin collides centrally with a coin of a larger nominal value (and heavier), it is bounced back and the target moves forward. In the last case, when a 10-cent coin collides with a coin of a smaller nominal value (and lighter), after the collision they both move in the direction of projectile movement before the collision. The guidelines also suggested that students should be familiarized with some extensions of this basic experiment: First, the projectile coin should remain the same and the target coin should be replaced by coins of different nominal values (and masses, e.g., 2-, 5-, and 20-cent coins, 1- and 2-euro coins), and second, the target coin should remain the same and the projectile coin should be replaced by coins of different nominal values.

Another variant of the experiment was a Newton’s cradle built from coins (see Fig. 3). Placing several coins of the same or different nominal values in a row and flicking a projectile at this coin chain again led to qualitatively different outcomes even if all collisions within the coin chain were central.

The last variant of the experiment is shown in Fig. 4. The target consists of two coins, each with the same nominal value (mass) as the projectile. The collision of the projectile with the first coin in the target is central. The projectile coin transfers its momentum to the first coin in the target and stops at the position of collision, just like in the first experiments with central collisions. After the first collision, the first coin in the target starts moving and immediately collides with the second coin in the target. This second collision is non-central, and after the collision both coins move in directions that are not the same and also different from the direction in which the projectile was originally moving. The second coin in the target is pushed and starts to move in the direction along the line connecting the centers of the target coins, as shown in Fig. 4 (a torque impulse during the collision obviously does not cause a significant change in the direction of motion). Even in such a simple experiment, one can easily observe a peculiarity of elastic, non-central collision of objects of the same mass: After the collision they move in directions perpendicular to each other.^{1,3,4}

## Collisional situations

Central and non-central collisions differ qualitatively in their outcomes, and the final positions of the coins serve as evidence of the type of the collision. If the experiments are performed carefully, students should recognize the following facts, patterns, and rules, which are summarized in Table I.

. | A) Central Collision . | B) Non-central Collision . |
---|---|---|

1 | The target moves along the line, connecting the centers of projectile and target coin at the collision. | |

2 | If the speed of the projectile is greater, the coins make longer paths after the collision. | |

3 | If the mass of the target is greater, the displacement of the target is smaller. | |

4 | The final positions of the coins [shown in Fig. 5(b)] are aligned along the same line as the initial positions of the coins [shown in Fig. 5(a)]. | The final positions of the coins are not aligned along the same line as the initial positions of the coins [as shown in Fig. 5(d)]. |

5 | After collision, target moves ahead all times (provided its mass is not so large that static friction force prevents movement). | After the non-central collision of the two coins of the same nominal value, their directions of movement after the collision are perpendicular to each other [see Fig. 5(d)]. |

6 | The resulting velocity of the projectile is predictably determined by the mass ratio of projectile and target. If the mass of the projectile is smaller than the mass of the target, the projectile rebounds after the collision. If it is greater than the mass of the target, the projectile continues to move forward after the collision, but at a lower speed than before the collision. | |

7 | In special cases, when the masses are equal, the projectile stops on impact and transfers its entire momentum to the target. | |

8 | If a target consists of a chain of coins of the same nominal value, at collision the motion of the projectile is transferred to the last coin in the chain. | |

9 | If the target consists of more coins, the outcome of the experiment is pretty much the same as the outcome of series of collisions would be. |

. | A) Central Collision . | B) Non-central Collision . |
---|---|---|

1 | The target moves along the line, connecting the centers of projectile and target coin at the collision. | |

2 | If the speed of the projectile is greater, the coins make longer paths after the collision. | |

3 | If the mass of the target is greater, the displacement of the target is smaller. | |

4 | The final positions of the coins [shown in Fig. 5(b)] are aligned along the same line as the initial positions of the coins [shown in Fig. 5(a)]. | The final positions of the coins are not aligned along the same line as the initial positions of the coins [as shown in Fig. 5(d)]. |

5 | After collision, target moves ahead all times (provided its mass is not so large that static friction force prevents movement). | After the non-central collision of the two coins of the same nominal value, their directions of movement after the collision are perpendicular to each other [see Fig. 5(d)]. |

6 | The resulting velocity of the projectile is predictably determined by the mass ratio of projectile and target. If the mass of the projectile is smaller than the mass of the target, the projectile rebounds after the collision. If it is greater than the mass of the target, the projectile continues to move forward after the collision, but at a lower speed than before the collision. | |

7 | In special cases, when the masses are equal, the projectile stops on impact and transfers its entire momentum to the target. | |

8 | If a target consists of a chain of coins of the same nominal value, at collision the motion of the projectile is transferred to the last coin in the chain. | |

9 | If the target consists of more coins, the outcome of the experiment is pretty much the same as the outcome of series of collisions would be. |

## The problems

The competition problems relate to outcomes of experiments, facts, patterns, and correlations listed in Table I, which should have been observed. Within the problems, some of the objectives are addressed directly and others indirectly, but almost all are reflected in the competition problems. We are happiest when the problems do not ask directly about the outcomes, but check whether the recognition of patterns can be applied to somewhat different and also more complex situations. For example, we do not ask “Which of the target coins (10-cent, 20-cent, and 50-cent) is displaced most in the central collision with the projectile 1-cent coin, if the projectile has the same speed in all three cases before the collision?” (with reference to observation 3), and if we have a question with a figure showing different displacements of three coins with the same nominal value after the central collision, we do not ask “In which collision of the projectile 1-cent coin with the target 50-cent coin the speed of the projectile was highest?” (with reference to observation 2), but rather combine both elementary collision events into a more complex example of problem 1, where observations 2 and 3 should both be taken into account, since both the mass of the target and its displacement vary.

The solutions to given problems can be found at the end of this article.

### Problem 1

A 1-cent coin collided centrally with a 10-cent, 20-cent, and 50-cent coin. The initial (photos) and final positions (drawn circles) of target coins are shown in the figure (Fig. 6). In which case was the speed of the 1-cent coin before the collision the largest?

### Problem 2

A 1-cent coin collides centrally with a 50-cent coin. Which figure (Fig. 7) shows correctly the position of the coins after the collision?

### Problem 3

A 50-cent coin collides centrally with a 1-cent coin. Which figure (Fig. 8) shows correctly the position of the coins after the collision?

### Problem 4

A figure (Fig. 9) shows a central collision of the projectile 10-cent coin K1 with a target, composed of two 10-cent coins K2 and K3. For each of the three coins K1, K2, and K3, write down their final positions after the collision.

## Comments and conclusion

We gave a description of the guided and structured collision experiment with coins. Several variants of the experiment (central and non-central collisions, same or different masses of the projectile and the target, the target consisting of one or more coins) with qualitatively and quantitatively different and meaningful outcomes were given. These outcomes were described, and a list of possible observations and conclusions about the patterns, correlations, and rules was written. At the end a set of competition problems were given that test whether the students had achieved these results.

The phenomenon under consideration—elastic collisions of round objects—is a very common and typical topic in high school or the first year of university studies. There are many articles in which the authors deal with specific aspects of ball collisions, and only a few are listed in Refs. 1–6. Nevertheless, we think that there is still room for another one (this one). Two main qualities of our article are first the simplicity of the experiment; no special equipment is required and outcomes are easy to observe even for much younger students. Secondly the problems are not easy; in order to solve them successfully, students must apply higher cognitive processes than just remembering what happens in the basic variants of the collision experiment. They must combine more than one rule and be able to analyze the complex situation and see it as a sequence of simpler situations.

Through a science competition we reach not only the pupils but also the teachers. We provide materials that they can use even if their pupils do not take part in the competition. Many process-oriented goals of the science curriculum can be achieved in regular science lessons through activities like the one described here. Pupils make their own observations and experiments, perceive interrelationships, formulate hypotheses about cause-and-effect relationships, and test them systematically, to mention only a few.

## Solutions

### Problem 1

Solution: C. Greater displacement of the target means greater initial speed of the projectile; greater mass of the target also means greater initial speed of the projectile. Reference to observations 2 and 3.

### Problem 2

Solution: A. A light projectile bounces back after colliding with the heavy target. Reference to observations 6 and 7.

### Problem 3

Solution: D. After the collision with the light target, the heavy projectile moves further forward and stops closer to the position of the collision event than the target. Reference to observations 6 and 7.

### Problem 4

Solution: K1 stops at B, K2 stops at E, and K3 stops at I. The projectile K1 first collides centrally with the coin K2 of equal mass and stops immediately after the collision at B, while K2 starts moving and immediately collides non-centrally with K3. After this second collision, K3 moves in the direction of the line connecting the centers of K2 and K3 at the moment of collision, in direction towards I, and K2 moves in the perpendicular direction to E. Reference to observations 1, 4A, 4B, 5B, 7, and 9.

## References

**Faculty of Education, University of Ljubljana, Slovenia**