A Cool Sport Full of Physics

Contents

• BODY OF ARTICLE
• Dissecting the Slapshot
• Colliding Players
• Ice Is Slippery, But Why?
• REFERENCES
• About the Author
• FIGURES
• TABLES

Of all sports, ice hockey is possibly the one with the widest array of physics elements in it. The game provides many examples that can bring physics to life in the classroom. Ice hockey (or just “hockey” as many Canadians would say) sees athletes sliding on ice at high speeds and in various ways, shooting and slapping pucks, and colliding against each other. The interaction between the skate blade and the ice is a problem of great physical complexity. The question “Why is ice so slippery?” has puzzled generations of scientists and, surprisingly, clear answers have come relatively recently. There is even some optics involved in hockey: how many sports are watched behind tempered glass (or Plexiglas) windows? The optical and mechanical properties of these materials are worth a physics classroom discussion. In this paper, I will review a few topics discussed at length in my book The Physics of Hockey.^1,2 Interested readers may also find additional articles on our website.³

One of the most intriguing and spectacular events in hockey is the slapshot, a technique used by hockey players to send pucks flying at 160 km/h or more. But before looking into this, let's begin with a comparative summary of projectiles in sports. Table I sorts common projectiles by velocity.⁴ The golf ball (tee shot) tops the list at over 250 km/h and, not surprisingly, with only 46 grams of mass it is also the lightest one. If we want to know how much effort is exerted by an athlete throwing a ball, the kinetic energy it carries is a better indicator. Taking mass into account andcalculating the kinetic energy, a completely different picture emerges in Table II.

The top three projectiles are from specialty events where the aim is to send a projectile the furthest possible distance. They happen to be the heaviest and slowest projectiles, their inertia enabling the athlete to deliver more power. Baseball batting is next on the list, but one must consider that it is a combined effort by the pitcher and the batter as the ball partially bounces off the bat. (A baseball player could not bat the ball that fast if it were stationary.) Footballs and golf balls carry much less energy than soccer balls and hockey pucks.

What determines the kinetic energy of sport projectiles? The player's technique plays a role for sure, but an equally important factor is how much muscle mass participates to the motion. For example, throwing a football mostly involves one arm (the triceps especially) and one shoulder; in contrast, kicking a soccer ball involves large muscle groups in the legs and torso (it also helps if the kicker runs toward the ball). As a rough rule of thumb, one horsepower of peak power is delivered per 10 kg of muscle mass working. In hockey, the slapshot requires work form the player's legs, torso, arms, and shoulders, hence the significant amount of energy delivered to the puck.

The slapshot is a violent collision between the hockey stick and the puck that happens in three stages. First, the upper body winds up and begins an accelerated rotation of the torso, shoulder, arms, and the stick. Next, the stick blade comes in contact with the puck and the ice, causing the stick shaft to bend and accumulate potential energy, as Fig. 1 shows. In the last stage, the puck accelerates and leaves the stick blade as the shaft returns to its original shape.

Figure 1.

To conceive a simple physical model of the slapshot, we may assume it to be an elastic collision between a rotating body (the hockey player and his stick) and a light stationary mass (the puck). Figure 2(a) illustrates the idea. In reality, some energy is dissipated in the process, but, as Fig. 3 suggests, there is plenty of elasticity in the stick to make it a fair approximation.

Figure 2.

Figure 3.

Newtonian mechanics is now used to find the puck velocity as a function of the problem's parameters. Conservation of angular momentum requires that

where I, _i, and _f are the moment of inertia of the player/stick system and its angular velocities just before and after the collision, respectively. The puck of mass m lies at a distance r of the axis of rotation of the player and leaves with a velocity v.

Let's assume the initial angular velocity and moment of inertia of the player/stick system are known. With Eq. (1), we are left with two unknowns, namely v and _f. Conservation of energy provides the necessary extra equation. Using

the kinetic energy of a rotating body, the following relation is obtained

We now put together Eqs. (1) and (2) to find the final velocity of the puck:

Given the biomechanical complexity of the slapshot, this formula may not have much predictive power in reality, but, as is often the case in physics, we can still learn a number of important lessons from it. Most obvious is the fact that the puck speed is proportional to the spinning velocity of the player. This makes intuitive sense. Second, the larger I is, the faster the slapshot, and this is where body mass and positioning technique come into play. Finally and most importantly, in the limit when I mr², which is often the case, the puck reaches a maximum velocity of v = 2 _i r. This speed limit is similar to what occurs in linear collisions: when a heavy mass collides with a light stationary mass, the latter will never acquire more than twice the velocity of the oncoming mass. A good example is a golf ball sitting on a tee hit by a club.

A 160 km/h slapshot would be difficult or impossible to attain from a direct elastic collision between the stick and the puck. Indeed, given a player's mass, stick length, and other parameters, the spinning velocity would be too high. So how do players do it anyway? The trick is to involve the ice. Watching a slow-motion replay of a hard slapshot, one can notice the stick touching and sweeping the ice before the puck. Hockey coaches teach young players—perhaps without knowing why—to hit the ice about 30 cm before the puck. The result is extra bending of the stick and some more oomph to the puck. To understand why, let's return to our model and suppose that instead of hitting the puck directly, the stick first hits a wall, as Fig. 2(b) shows. The rotation energy of the player is then gradually stored in the flexing of the stick until the player stops turning altogether. If the wall is then removed and the puck put in contact with the stick, the puck can now receive a great deal more speed. The puck velocity is then limited by the energy of the system:

which is typically much higher than the limit v 2_ir.

A similar effect occurs during a wrist shot. The player often pushes against the ice and then lets the bent stick do the work against the puck.

Hockey is the fastest contact sport on Earth. Starting from rest, NHL players can complete a rink lap in about 14 seconds, clocking an average velocity of 40 km/h. Although they rarely happen at maximum speeds, collisions and body checks deliver significant impact forces. The kinetic energy dissipated in a collision is an important factor determining the impact force and the chances of injury.

The theorem of kinetic energy says that the work done by a net force F on a body is equal to the variation in kinetic energy F. Expressed mathematically in one dimension:

where is the average net force applied over a distance x. This equation is useful to estimate the force of impact, = K/ x, between colliding players. Because kinetic energy goes as the velocity squared, speed is more important than body mass. Thus, an average-sized hockey player (200 lbs) skating at 40 km/h carries as much energy as a heavy football linebacker (350 lbs) running at 30 km/h.

Let's consider the mid-ice collision depicted in Fig. 4. Let's suppose the two players each have a mass of 90 kg and are skating at a moderate speed of 8 m/s. They then carry together a total of 6000 J, or enough to power an average light bulb for a minute and a half. After a collision, the final velocity is relatively small, so one may assume that all the energy is dissipated. Taking x = 20 cm as an estimate of the deformation length of each player (and their protective gear), the average force of impact is 14 kN. The impact force is severe but doesn't last very long.

Figure 4.

Liquid water has a unique combination of physical and chemical properties that make life possible, among other things. But ice is equally amazing. For example, with a friction coefficient as low as µ = 0.005, it is one of the most slippery substances we know.⁵ Few contenders can beat that, synovial joints being one of them. But ice behaves trickily too. For example, water expands upon freezing, unlike most materials; Put two ice cubes against each other and soon they won't slip, they will bond solidly.

A popular explanation for why ice is slippery invokes the melting of ice when it is put under pressure.⁶ At 0°C and 1 atmosphere of pressure, the melting point of ice drops by 1° for each 130 atm of added pressure. A neat experiment to show this effect is to connect two masses with a thin wire and suspend them around a block of ice supported at both ends. The wire will slowly cut through the ice while leaving it intact. The water squeezed under the wire melts, and the water moves to the top, where it freezes instantly for lack of pressure. As a result, the wire magically migrates through the cube! The process, called “regelation,” was discovered by Michael Faraday. Can the same pressure melting explain why hockey players slide so easily on the ice? In principle it is possible, but the melting point of ice squeezed under the skate of the heaviest player drops by a fraction of a degree only. It might become a factor near the 0°C mark, but it does not explain why skating is possible at −20°C let alone explain why something as light as a hockey puck can slide just as well.

Another popular but equally faulty argument invokes the heat that is generated when two surfaces rub against each other. Frictional melting of ice, as it is called, can also play a role in creating a thin layer of water between the skate and the ice, but again the temperature rise is minute. The answer must lie elsewhere.

Techniques such as electron scattering, nuclear magnetic resonance, and atomic force microscopy have revealed important secrets about molecules lying at the surface of ice.⁷ The crystalline structure of bulk ice is hexagonal, like the graphite in your pencil, with loosely tied sheets of molecules. At the surface, however, water molecules are not so orderly. With fewer neighbors to bond with they tend to move around, forming a liquid-like layer. This wet layer is very thin—nanometers deep—but it seems to play an important role in making ice slippery. It exists without adding pressure or rubbing, even at temperatures as low as −200°C. Ice is, one could say, slippery all by itself. Below this thin layer, the “dry ice” provides as much friction as many common materials like plastics.⁸

It should be pointed out that the wet layer phenomenon is not unique to ice — it has been observed on some metals and semiconductors—but the reasons why it works so well for ice are not fully understood. Although we teach friction as part of every introductory mechanics courses, it is a complex process that depends on many factors, including, for the case of ice, the presence of chemical impurities.

References

1. A. Haché, The Physics of Hockey (Johns Hopkins University Press, Baltimore, 2002). Available through AAPT. first citation in article
   
2. John L. Hubisz, “Book Review: The Physics of Hockey,” Phys. Teach. 45, 59 (Jan. 2007). first citation in article
   
3. See http://www.thephysicsofhockey.com first citation in article
   
4. The Physics of Sports, edited by A. Armenti (Springer Verlag, New York, 1992). first citation in article
   
5. J. J. de Koning, G. de Groot, and G. J. van Ingen Schenau, “Ice friction during speed skating,” J. Biomech. 25, 565–571 (June 1992). first citation in article
   
6. S. C. Colbeck, “Pressure melting and ice skating,” Am. J. Phys. 63, 888–890 (Oct. 1995). [ISI] first citation in article
   
7. R. Rosenberg, “Why is ice slippery?” Phys. Today 58, 50–55 (Dec. 2005). first citation in article
   
8. H. Bluhm, T. Inoue, and M. Salmeron, “Friction of ice measured using lateral force microscopy,” Phys. Rev. B, 61, 7760 (March 15, 2000). [ISI] first citation in article
   

Alain Haché, after receiving his PhD in quantum optics at the University of Toronto in 1997, took an academic appointment at Université de Moncton, his alma mater, where he's now Canada Research Chair in Photonics. His decidedly amateur hockey career as a goaltender began when he was 5 years old and it will continue, he hopes, for many more years.alain.hache@umoncton.ca.

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Fig. 1. Michael Peca (formerly of the Buffalo Sabres) delivers a hard slapshot. The photo shows the importance of stick flexibility, as it can bend by as much as 30 degrees. First citation in article

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Fig. 2. a) A simple model of the slapshot assumes a direct elastic collision between the puck and the rotating player/stick system. b) Players seldom hit the puck directly, however, but instead they flex the stick against the ice before touching the puck (the ice is schematized by a fixed wall). Greater puck velocities are achieved this way. First citation in article

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Fig. 3. Al MacInnis on his way to capturing another title at the 2000 NHL Hardest Shot Competition. First citation in article

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Fig. 4. Frontal collisions, such as this one between Kris Draper and Pavel Bure, tend to be rare but also severe. First citation in article

Table I. Top speed of common sport projectiles.
Projectile	Top speed (km/h)
Golf ball	255
Tennis ball	225
Baseball (batting)	190
Hockey puck	170
Baseball (pitching)	160
Javelin throw	110
Soccer ball (kick)	105
Discus throw	90
American football (throwing)	70
Shot put	50

First citation in article

Table II. Common sport projectiles sorted by maximum kinetic energy.
Projectile	Mass (g)	Top Speed (m/s)	Energy (J)
Shot put	7260	15	780
Discus throw	2000	24	600
Javelin throw	800	31	390
Baseball (batting)	145	53	205
Soccer ball (kick)	430	30	195
Hockey puck	170	47	185
Baseball (pitching)	145	44	145
Golf ball	46	71	115
Tennis ball	57	62	110
Football (throwing)	400	20	80

First citation in article