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The Physics Of Hockey

picture of ice hockey

Source: http://www.flickr.com/photos/cubesbyamanda/2385836783

The physics of hockey is a broad subject of analysis, covering key aspects related to performance and equipment design. Experienced hockey players are generally aware of (either directly or indirectly) how physics plays a role in their ability to play the game. The purpose of this page is to discuss how the game of hockey and physics go together.

Note that the information presented below is largely based on the book by Alain Haché: The Physics of Hockey, and the website: http://www.thephysicsofhockey.com

The book by Alain Haché goes into much greater detail on the physics of hockey than I do on this page.

Physics Of Hockey – Skating

When skating, the skates of a hockey player do two things: They glide over the ice and they push off the ice with the edge, in order to gain speed. Experienced hockey players make this combination of movements look like second nature, which is to be expected since they must be able to play offensive and defensive positions, while also controlling the puck.

The physical properties of ice is what allows hockey players to maneuver the way they do. For instance, the low friction of the skate blade with the ice is what allows a hockey player to easily glide over its surface. And the physical make up of the ice is what allows a player to dig in with his skate in order to go around a turn, speed up, or stop.

A hockey player propels himself forward by pushing off the ice with a force perpendicular to the skate blade. Since the friction of the blade with the ice is almost zero, this is the only way he can propel himself forward. The figure below illustrates the physics behind this principle.

schematic of hockey player pushing off the ice

As the hockey player pushes off with his rear leg, a perpendicular force F is exerted on the skate by the ice. The component of the force F that points forward (in the direction of motion) is what pushes the player forward. At the same time, his other skate is either raised or gliding on the ice. As he moves forward he then switches to the other leg and pushes off the ice with that one, and the process is mirrored. To push off the ice with greater forward force (and accelerate faster), the skater increases the angle α, which increases the component of force in the direction of motion.

To avoid turning their backs on the opposing team, hockey players sometimes skate backwards using a gliding pattern in the shape of a lazy "S" (as shown below). In this skating pattern, the player's blades never leave the ice. However, the player cannot push off against the ice as hard as he does when skating forward, which means he cannot go as fast.

schematic of hockey player pushing off the ice and skating backward

In this technique, the player pushes against the ice with his push-skate facing inward, while his other skate glides. As he moves backwards he then switches to the other leg and pushes off the ice with that one, and the process is mirrored. Thus, the physics of skating backward is similar to the physics of skating forward.

A hockey player can at most move his feet at about 7 m/s, and the greatest forward push force will be when he begins skating from rest. At this point the velocity of his foot relative to the ice is 7 m/s. As the player gains speed this relative velocity changes. For example, if he reaches a speed of 5 m/s, the relative velocity of his foot relative to the ice is 2 m/s (assuming he moves his leg backwards, with no sideways component of velocity), and the push force is less as a result. Consequently, there is a maximum speed a hockey player can reach, which is directly influenced by how fast he can move his feet on the ice. However, the maximum speed the player can reach is not necessarily 7 m/s. It can be much more than this if the player, when pushing off the ice, moves his leg backward with a sideways component of velocity. To understand this, and to determine the maximum possible speed which can be reached, we must look at the biomechanics of the player on the ice. The biomechanics of a player as he moves on the ice is another useful analysis in the physics of hockey.

To maintain his balance when accelerating forward, a hockey player will crouch forward in the direction of motion. This prevents him from falling (tipping) backwards due to the torque caused by the forward component of the force F. By crouching (or bending) forward, the player is moving his center of mass forward which creates a counter-torque. This counter-torque balances the torque caused by the forward component of F, and this prevents him from falling (tipping) backwards.

The design of the hockey skate is another important factor related to the physics of hockey. A hockey player's blades must be able to support his quick acceleration, turns, and stops. This is accomplished by grinding a slight hollow into the bottom of the blade. This creates two sharp edges which "bite" into the ice, and prevent slipping. The figure below illustrates this.

diagram of hockey players skate blade

Due to heavy use during a typical game, a player's blades must be regularly kept sharpened in order to maintain optimal performance. If the edges become dull the result can be a player's foot slipping out from under him as he goes around a turn or attempts to stop.

The Puck

A hockey puck is made of a hard vulcanized rubber material, able to withstand the high level of wear and tear during a game. They are colored black in order to be highly visible against the surface of the ice.

Hockey pucks are frozen prior to being used in a game. This reduces the level of friction the puck has with the ice and allows it to travel further on the ice, without "sticking". This is convenient from a player's point of view since he prefers to maintain his momentum on the ice without having to stop and hit the puck again to get it moving.

Freezing the puck is also done to intentionally reduce how much it bounces during play. This enables better control of puck movement.

The Hockey Stick

The hockey stick has several different features. It is designed to enable good puck control, while also being lightweight and strong enough to withstand the stresses placed on it during use.

One of the key features of a hockey stick that affects puck control is the curvature of the blade, which acts as a type of self-centering mechanism. When the puck is struck the curvature of the blade "forces" it towards the bottom of the curve, where it tends to sit for the brief duration of impact before flying off. This allows a player to make more consistent shots since the puck tends to fly off the same part of the blade every time.

However, for the sake of fairness and uniformity of play, regulations typically limit the amount of curvature a player's blade can have. For example, in the NHL the maximum curvature is defined as follows: The perpendicular distance from a line drawn from the heel of the blade to the end of the blade, and to the point on the blade of maximum distance, shall not exceed three-quarters of an inch. A diagram of this is shown below.

diagram of hockey stick showing maximum curvature of blade

In the above diagram the maximum allowable distance denoted by the blue line is three-quarters of an inch.

Some players curve their blades closer to the end of the blade, which makes it a bit easier to scoop the puck away from another player. Personal preference is a main factor in how players curve their sticks.

Curvature of the blade also allows players to more easily put spin on the puck which gives it gyroscopic stability during flight. This makes it more likely that it will land flat on the ice. Applying tape to the blade improves friction between puck and blade. This aids the ability to put spin on the puck.

Another feature of a hockey stick that affects puck control is the "loft" or "face" of the blade. This is the tilt angle of the blade, visible when looking at the stick from directly above. This is illustrated in the figure below.

diagram of hockey stick showing tilt angle of blade

A greater tilt angle makes it easier for a player to lift up the puck and get it airborne. Again, how much tilt works best comes down to the personal preference of the hockey player.

A feature built into hockey sticks, tailored to a player's style of play, is the angle of "lie". This is the angle the blade makes with the shaft. This is represented by the angle θ in the figure below.

diagram of hockey stick showing angle of lie

Players usually seek a lie angle that will put their blade flat on the ice while they are in their typical skating stance.

As shown in the figure above, the toe is the very end of the blade. The toe comes in two basic shapes: round and square. The difference between the two is that the round toe allows more ability to control the puck at the tip, while the square toe increases the blocking area at the tip.

A good choice of material for hockey sticks is carbon fiber. It is lightweight and has high strength. This is important for ease of puck control and for making shots, such as the slapshot (which will be discussed next). This is a good example of how material science is an important part of the physics of hockey.

The Slapshot

In the slapshot, players can clock puck speeds of over 100 miles per hour, making it the hardest shot in hockey.

The hockey player begins the slapshot by raising the stick behind his body, as shown below.

picture of hockey player beginning the slapshot

Source: http://en.wikipedia.org/wiki/Slapshot. Author: http://www.flickr.com/photos/somegeekintn

Next, the player violently strikes the ice slightly behind the puck, and uses his weight to bend the stick, storing energy in it like a spring. When the face of the blade strikes the puck the player rotates his wrists and shifts his weight in order to release this stored energy and transfer it to the puck. The result is the puck reaching a speed faster than it would if the player simply hit the puck directly. The kinetic energy of the puck after impact is equal to the stored energy in the hockey stick.

The figure below shows the point of impact between the stick and puck. You can clearly see the bend in the stick.

picture of hockey player during the impact stage of slapshot

Source: http://www.flickr.com/photos/dinur/2952972685

Thus, the physics taking place here is the transfer of energy from player to stick, and from stick to puck. The advantage of storing energy in the stick is that (upon release) it strikes the puck faster than the player can, causing the puck to reach a greater speed.

The Goaltender

picture of goaltender

Source: http://www.flickr.com/photos/howlingmad/465762068

The role of the goaltender is to block shots made by the opposing team. To protect his body from injury he wears protective gear to absorb the impact of the puck with his body. Due to his high level of exposure to high-speed pucks he wears even more protective gear than the other players. However, the weight and bulk of the gear can slow down and restrict the movement of the goaltender somewhat. Thus, strength and cardiovascular training, as well as the learning of good technique and efficiency of movement, is an essential part of being a good goaltender.

The Lacrosse Style Goal

This is a particularly interesting subject in the physics of hockey. In the lacrosse style goal, the hockey player skillfully maneuvers the puck into the net, while maintaining contact between puck and blade. The picture below shows Mike Legg who, in 1996 (while playing for the University of Michigan), scores a lacrosse style goal from behind the net. The approximate trajectory of the puck from ice to net is represented by the blue curve.

picture of lacrosse style move in hockey

To begin the move, Mike Legg orients the puck on its edge so that it touches the blade of his stick head on. He then guides the puck along (using the blade of his stick) such that it follows a curved trajectory, as shown. This curved trajectory causes the puck to experience centripetal acceleration. The centripetal acceleration points towards the center of curvature of the curve, in the direction of the red arrows (shown in the picture). This centripetal acceleration in turn causes the puck to "push" against the blade hard enough so that it doesn't fall off due to gravity.

There are basically two things that must happen in order for this trick to work:

(1) He must move the puck along the trajectory at a high enough speed (v) to generate a high enough centripetal acceleration (ac), to generate sufficient contact force between puck and blade. (Note that ac = v2/R, where R is the radius of curvature along the trajectory).

(2) At the same time, he must orient the blade so that the contact side faces the center of curvature of the trajectory – in other words, it must face in the direction of the red arrows (shown in the picture). This allows the puck to "push" against the blade with enough contact force to avoid falling off due to gravity. This happens by way of the friction force between blade and puck. The friction force is proportional to the contact force. So a high enough contact force generates enough friction force to counteract the force of gravity pulling down on the puck.

The lacrosse style goal must be seen to be fully understood. To see this goal made by Mike Legg watch the video below.

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