Physics 208A. Volume 6.1
Last updated on 21 May/98
Common questions that arise in baseball and tennis regard choices of the bat or racket for most effectiveness. The choice would be a function of the size and skills of a particular player. The magazines have advice on this and tennis racket makers hype their products. Advice is given on how to grip the implement to have the most effectiveness.
Sometimes, the advice is contradictory so that the player is left to his own devices. Fortunately there is a lot of science on bats and tennis rackets and the magazine advice has become more uniform and also scientifically correct in the last decade. Some of the concerns have been in the realm of:
The job of the baseball hitter is to consistently hit the ball well. This could be singles that are hit at medium range but away from fielders. However, the games are often won with long balls hit by the powerful hitters. To hit a homerun over the nearby fences (near the foul lines) requires a ball to be launched with the velocity to have a range of about 90m ~ 300'. We can use our derived formula of Range(launch velocity) that we derived in chapter 4. Unfortunately, that formula is derived for the range for a ball in a vacuum. The baseballs are hit with high velocities where air resistance is an important factor and can cut the range down to about 1/2 of what is expected in a vacuum. The graph below shows the calculated range (launch velocity, launch angle = 40 deg).
The minimum for a home run is about 40 m/s or 88 mi/hr. This means that a good thrower like a pitcher can throw a ball over the outfield fence near the foul line.
For hitting, the batter obtains the launch speed by a collision between the pitched ball and the swung bat. The higher the speed of the pitch and of the bat results in a higher launch speed. For a given bat speed, the heavier the bat, the higher the launch speed. The formula for the launch speed as a function of the pitch, the bat speed, the coefficient of restitution (CoR) and the bat and ball speed is shown below.
Vlaunch = Mbat/(Mbat+Mball)*((Mball/Mbat - CoR)*(-Vball)+(1+CoR)*Vbat)
This has been derived from the law of conservation of momentum and using the measurement of the elasticity of the collision CoR = velocity out/velocity in. We can approximate this formula for a batter swing a 30 oz bat at the 5.1 oz baseball with the formula
Vlaunch = 0.5*Vpitch + 1.3Vbat.
The figure below illustrates the launch speed(bat speed) for bat weight = 30 oz and two pitch speeds.
It's clear that the higher the bat speed or pitch speed, the bigger the launch velocity. For a 60 mi/hr pitch, the smallest home run bat speed is about 55 mi/hr. For a 90 mi/hr pitch, the necessary bat speed is much less at 45 mi/hr.
A given ball player will be able to swing a lighter bat with a higher bat speed given the Newtonian relationship of force and mass. However, the smaller mass bat will rebound more from the collision so won't deliver as much of a launch speed for a given bat speed. These are contrary effects. However, we can come to some conclusion about these effects by using measured bats speeds (bat weight) and the calculated launch speeds (bat speed , bat weight) to calculate the range of the hit ball. The following graph shows the bat speed obtained by little league players and by professional players.
The relation of bat speed and weight are not accurately calculable because the force that a batter can apply varies as the speed so that we cannot just scale the batspeed at one weight by a 1/m relation. It's clear that higher bat speeds are obtained at smaller bat weights. We can try this out in class using the bat speed meter and various simulated bats.
The bat speed can be converted to a launch speed (pitch speed). The figure below illustrates the launch speed (bat weight) for a fixed 90 mi/hr pitch.
The launch speed varies as expected and a home run can be obtained by a professional player swinging a bat as light as 25 oz. However, the launch speed continues to be bigger for heavier bats but it plateaus at about 30 oz. There is no advantage for using heavier bats. There could be a serious disadantage with the heavier bats given that the time to swing the bat increases with the heavier bats. As we have already noted, the batter only has about .46 seconds to recognize the pitch and then to accelerate the bat up to speed in the target area. The latter will take only about 0.25 seconds. The heavier the bat, the earlier he must begin his swing and the less time the batter has for sizing up the ball.
While Babe Ruth used a bat of about 50 oz, the modern long ball hitters like Ken Griffey Jr of the Mariners uses a bat of about 30 oz. A 36 oz bat is considered a heavy bat today. The bats are shaped to give a big target area so the barrel is big while the handle is quite slender. This makes them vulnerable to breakage. [The rules of major league baseball requires that wooden bats be used.] The mass being in the barrel makes most of the total bat mass be effective in the hit. The springiness of the thin handles is not detrimental when the ball is hit in the center of percussion.* When the ball is hit at the center of percussion, the force of the hit does not exert ANY force on the hands. This is the SWEET SPOT. In this case, the bat acts as if it were free since the struck ball exerts no force on the hands, the hands can exert no force on the ball. A bat held with a relaxed grip is the most effective way of holding the bat. Getting the arm or the body behind the hit has no meaning. Once you've launched the bat at the ball, the bat acts on its own just as well even if you were not holding the bat at all at the moment of impact.
A ball struck away from the center of percussion will vibrate causing the stinging of the hands and may break the bat. In the case of the broken bat hit, the ball will have a poor launch velocity because the effective mass of the bat is smaller. In the case of the ball being hit closer to the handle, the hand force at the time of the hit may have some influence on the launch speed. Since the batter will sometimes hit the ball at some place other than at the center of percussion, it would be well to have a firm grip on the bat to add some effective mass. The firm grip may help provide an accurate swing of the bat and of course, you must not let the bat slip out of your hands.
Having a firm grip doesn't mean grabbing the bat handle with a tight grip. Over gripping the bat is detrimental because the tightness in the grip is likely to cause tightness in other joints which would be detrimental for obtaining the needed high bat velocities. The tightness (contraction of the antagonists) will impede the bat speed and so should be avoided. The major league players often wear sticky gloves and have pine tar on the bat to help provide a firm grip without tight muscles.
The follow through of the bat has no influence on the launch speed as the collision occurs over a very short time (one millisecond) so that the ball is well away from the bat very quickly. In the one millisecond of the contact, the 55 mi/hr bat would have moved only by 2 cm. However, the coaching advice to hit through and follow through are correct because this well set the batter up to provide the proper bat velocity during the one millisecond of the collision of bat and ball.
As mentioned earlier, the desireable wooden bats with the thin handles are rather fragile and with bats costing >$100 each, the costs of bats is rather high and not affordable by most teams outside of the majors. Modern technology has given us the aluminum bat. The material is very strong and the bats can be made much lighter while still mainting a large target area. Besides having the proper diameter barrel, the distribution of the mass in the aluminum bat is made so that the sweet spot of the bat is made significantly larger than the sweet spot of a wooden bat. The present day aluminum bat has a sweet spot that extends farther toward the handle. For this reason the inside pitches can be hit much better with an aluminum bat compared to the wooden bat. Players going up from college or the minors to the major leagues find that the decreased ability to hit inside pitches one of the more difficult challenges to overcome. The aluminum bats are a little more elastic than wooden bats but the elasticity of either bat does not play a large role in baseball. The inelasticity of the baseball itself dominates the collision.
There was a period in which illegal corked bats were used with the idea of producing longer hits. The practioners believed that they had a higher CoR and also gave higher bat speed. The bats were made by making a large axial hole in the bat and then filling the hole with super balls or with cork. These bats gave the batter an advantage. The commisioner of the National League appointed a Physicist of the National League, Robert Adair, to investigate this. The investigation found that the CoR was not improved The wood is so stiff that the materials in the cavity had no effect but to damp out the characteristic sound of a hollow and illegal bat. The modifications to the bat made them somewhat lighter and with a bigger sweet spot so that they were almost as good as the best aluminum bats. Today in the major leagues, the bats are made with a hollow in the far end of the bat which makes them lighter and they are almost as good as the corked bats of yesteryear.
Robert Adair, a particle physicist at Yale University wrote a book, The Physics of Baseball, out of his experience and subsequent investigations. The book makes for many hours of enjoyable study.
The comments made about baseball hitting apply in tennis. The differences we must make is that the racket strings provide a trampoline effect for the collision making the CoR for racket-tennis ball higher than the CoR for baseball. The tennis ball as tested by bouncing on a hard floor has a CoR of only 0.5 at 55 mi/hr so is lower than the baseball at the same speed. The energy lost comes from the distortion of the tennis ball where the energy is not recovered on the rebound. However, the strings of a tennis racket are usually quite elastic. A ball will bounce off the strings of a racket with higher CoR than off a hard surface. The reason for this is that the strings cuishon the collision which now takes place over 5 milliseconds [compared to 1 or less milliseconds for the baseball collision]. Since F = Dp/Dt, for the same Dp, the force is much reduced and the ball is distorted less with the consequence of less energy loss by the collision. In order to enhance the trampoline effect, the strings should be as loose as possible as compatible wit ball directional control. [This is contrary to the instinctive idea that the strings should be tight for high power.] Gut strings have more elasticity than nylon strings and so are often used by high level players.
The invention of the big racket by Prince a couple of decades ago has made a big difference in the game. The sweet spot of the racket was enlarged by a large amount with this design and was also made to be where the racket has a larger effective mass. This gives the player much more freedom to have high racket speed with the subsequent loss in spatial accuracy and still be confident of hitting in the sweet spot. Since the sweet spot is enlarged, the fraction of hits which away from the center of percussion is reduced so that the fatiguing and injury [tennis elbow] from mishits causing shock and vibrations which are transmitted to the hand and to the elbow are much reduced. This has extended the life of many tennis players. The design of the heads of the big rackets made the sweet spot in the middle of the strings so that accidental hits on the frame became much reduced in frequency. The consequence is that in the good hit, the strings are stretched symmetrically in the big rackets so that directional control is improved.
Since the contact time is much longer, the ball stays on the strings for 5 milliseconds so that a racket swung with a speed of 20m/s will travel 10 cm. Follow through has a little more meaning here than in the baseball case. I don't know that the rolling over over the racket to produce top spin as advocated by some coaches can be done in this time span. The top spin is most easily produced by the swiping of the ball by adding an upward velocity to the racket where the 5 milliseconds of contact time gives ample opportunity to give a lot of spin to the ball.
The relaltion of racket and input speed to the launch speed of tennis is very similar to the formula given above. In this case, the higher CoR is offset by the smaller striking mass of the racket. we have:
Vlaunch = 1.5 Vracket + 0.5 Vball
A player receiving a service doesn't have much racket speed to hit a devastating return. Andrei Agassi is the master of this. He stands close to the baseline and uses a very short backswing so that he doesn't develop so much racket speed but he hits accurately on the sweet spot. You might think that after several returns, the ball will reach lethal speeds by building energy with each hit! This doesn't happen because the air resistance is quite big so that the ball is slowed considerably between hits.
The end of the racket head farthest from the handle has a really low CoR. If you want to disguise a drop shot, a hit on the part of the strings will give a very slow launch speed. Since your opponent has to cue on your motion, he will gauge that the ball has a bigger velocity than what you obtained and hence gives you an advantage at winning the point.
The tennis player is normally not as challenged as the baseball player in hitting on the sweet spot. Normally, a good player will have a small dark spot on the sweet spot of the racket [no marks on the frame!]. This means that the grip of the racket can be quite loose with no loss in launch velocity with the subsequent improvement in the swing. The loose grip is also the best way of dissipating the vibrations that result from a mishit so that your elbow is not punished for this act. The grips on rackets has improved with the use of the sticky rubbery coverings and the slight enlargement of the end of the handle to give you a relaxed grip with no fear of the racket slipping in your hand.
Soccer and Football.
The kicking of a football or soccer are sports where the science of collisions is important. The principles discussed above hold true here as well. The biggest launch speeds result from hitting with a big foot speed and with a big effective mass. Good Kickers will have excellent flexibility and powerful extensors to accelerate the foot to a high speed at the collision point. The soccer type kick has surpassed the old football place kick with a heavy boot tip. In the soccer kick, the instep is the striking point rather than the shoe tip. Striking with the instep may give a higher foot velocity. The main effect in getting a higher launch velocity is the increase in the striking mass by using the instep. The boot is closely pressed to the big bones in ones foot so that the large part of the foot is the striking mass. In the toe kick, the toe of the boot is connected rather loosely to the rest of the foot so that the striking mass is effectively only a small part of the foot's mass. Remember, the bigger the striking mass, the higher the launch velocity. Some kickers have even discarded the use of boots altogether and will place kick with a bare foot but with the soccer type instep kick... not the toes.
Hitting in martial arts.
As we had earlier discussed in the history of physics of sports, the idea of hitting in martial arts is that the high speed and use of a large and hard hitting mass will cause localized damage and great pain to the adversary if the targets are well chosen. Using the bones of the side of the hand gives a large striking mass and which is quite robust. This is in contrast to the more usual western boxing style which uses the front of the fist which is somewhat more elastic and also more prone to injury.
In the eastern martial arts, the use of lightweight batons for hitting is chosen rather than the use of heavier baseball bat type weapons favoured by western goons. The higher speeds obtained by the lightweight batons cause local deformations and high pain without the severe injuries caused by the heavy weapons. It's for this reason that most riot policeman in the world today use lightweight batons when needed. The purpose is to immobilize the perpetuators without causing serious injury when used correctly.
Use of hand weapons like swords also follow the physics of bats. To have a high degree of effectiveness, a high blade speed is obtained with a strike at the target on the sweet spot of the sword to obtain clean through cuts heavy materials. [I try to use my machete this way on my bushes and it works very well. After a lot of cutting, if I'm good with using the sweet spot, there is little hurt in the wrists and elbows from the shock of the strikes.]
The western boxing hits with gloves does not cause localized damage to the target areas but when applied to the head, will accelerate the head backwards or sideways. The collision time is much longer than the martial arts blow so that a larger part of the hitter is involved to give a large striking mass. A fighter who is tired and hurt will be unable to keep his head from being snapped away from the hits. A good referee will stop the fight to keep lethal damage from being done by this act. Why is this kind of hitting so damaging?
The brains in your head is a gooey blob that sets in your skull. It has inertia and so would remain at rest. When the skull is accelerated, the brain stays still and the hard skull collides with the side of the brain. This causes lesions in the brain.. a concussion. Sometimes this causes death almost immediately. In more common cases, the lesions in the brain accumulate and the brain function is impaired... sometimes severely. "Punch drunk".
This kind of collision of skull with brain occurs in many other cases. When a linebacker body slams a quarterback, the quarterbacks head snaps back and a concussion and some brain lesions occur. The rate of serious brain injury to quarterbacks in the NFL has been increasing and the NFL administrations are looking for ways of reducing this without reducing the nature of the competitiveness and entertainment value. It seems that the dangers are known to the players but they have chosen a Faustian bargain.
Adults angrily quelling a crying infant have been known to kill infants inadvertantly by shaking the infant. The head of the infant [or child] snaps back and forth with the subsequent collision of skull and brain causing many brain lesions. The shaken infant death syndrome has sadly been added to our lexicon.