The Mysteries of a Baseball Bat

Transcription

1 Team # 804 Page of 3 The Mysteries of a Baseball Bat Abstract The construction of the bat is significant in a baseball game, certain hit point and certain bat material may cause a quite different result. To inestigate the inner physics of a baseball bat, we analyze arious configurations of the collision process. In finding the sweet point, we confirm that the sweet spot is in the fat part of a bat in a logical way. We also get a result that mental bat has a larger sweet spot zone than a wooden bat. In soling the second and third question, we focus on two most important aspects on which the problems depend: mass and elasticity. In the respect of mass, we construct a model by using the momentum conersation; we get a conclusion that the mass of the bat affects the speed of the ball after hitting. While in the respect of elasticity, we construct another model by using the physics theories such as the momentum theory and the energy conersion, etc. In the third problem, we construct a similar model and use the trampoline effect to analyze the issue. What s more, though the models of the two problems in elasticity are similar, but their mechanism is quite different. Adding some other information and analysis such as the speed of the ball and the size of the sweet spot zone, we get the conclusion that using the corking bat or the metal bat can both make the ball fly farther and faster after hitting, thus help the player make good achieement. For a competition aiming at safe and fair, the corking should be limited and the mental bat should be prohibited.

2 Team # 804 Page of 3 Contents Introduction 3 Notations and Definition 4 3 Simplifying Assumptions 4 4 The Model 5 4. Finding the sweet spot 4.. Illustration of the sweet spot 4.. A ball-bat collision model 4. Corking 4.. Mass effect 4.. Elasticity effect 4.3 Wood or Metal 4.3. Mass effect 4.3. Elasticity effect Other factors concerned 5 Comparisons between Corking and Metal bat 9 6 Adantages and disadantages of the model 9 7 Further discussions 0 8 References 3

3 Team # 804 Page 3 of 3 Introduction As one of the most popular games in America, Japan and some other countries in the world, baseball has been drawing people s attention in ariety of ways. Let alone its strong effect of bringing people in different parts of the country together because of the passionate in a same player or team, baseball has gie its loers a space and a chance to study the physics in hitting a ball as well as the features a bat may has ary from the location of the so called sweet point to the ariation of the hitting effect associated with material and inner construction of a bat. A sweet spot on a baseball bat is a physical place where a combination of factors suggests a particularly suitable solution that maximum power is transferred to the ball when hit. Wheneer the player hit the ball on the sweet spot, the hit will be the most powerful and cleanest one compared with any hit on the other part of the bat. Howeer, the sweet spot is aried based on the difference between bats, which means that this physical feature of the bat may ask eery good hitter to be a physician. As time goes by, the kinds of materials made from bats are increasing. As it is for now, aluminum bats are popular between amateurs while professional players can only use wood bats in the games for metal bats are forbidden by the rules. The majority of wood baseball bats today are made from northern white ash harested from Pennsylania or New York. White ash is used because of its hardness, durability, strength, weight and "feel". Trees that proide the lumber for baseball bats are often 50 years old, and of all the lumber harested - the top 0% is saed for pro bats. The introduction of aluminum baseball bats in the 970's foreer changed the game of baseball at eery leel but the pros. Despite general high cost than any other materials, aluminum bats are lighter in weight, which increases control and bat speed, mean while, they are stronger and can hit a baseball significantly further than wooden bats. In baseball, a corked bat is a specially modified baseball bat that has been hollowed a cylinder on the top and filled with cork or similar light, less dense substances to make the bat lighter without losing much power. A lighter bat gies a hitter a quicker swing and may improe the hitter's timing. Howeer, since the bat is lighter, the ball does not necessarily trael as far as with a heaier bat, but usually only by a few feet at most. In Major League Baseball, the size of the hole is limited by the rule, the depth should be no more than.5 cm and the diameter should be within.5 cm to 5. cm.

4 Team # 804 Page 4 of 3 Notations and Definition Symbols Notes m The mass of the baseball M The mass of the baseball bat r ball, i The elocity of the ball in the time interal i r bat, i The elocity of the bat in the time interal i r I The ibration impulse on the hitter s hands α The angle between the ball and the bat when contacting β The angle between the ball and the bat when dissipating J The moment of inertia of the bat ω The moment of elocity of the bat L z The moment of rotation about z-axis ω The moment of elocity of the bat in the time interal i e A bat, i The collision efficiency The relatie speed of bat to the baseball bat, f r The ratio m M e The coefficient of restitution m The mass of the ball in the figure 5 m The mass of the cube marked in the figure 5 m The mass of the cube marked in the figure 5 3 The elocity of ball before the impact as is marked in the figure 5 The elocity of cube before the impact as is marked in the figure 5 The elocity of cube before the impact as is marked in the figure 3 5 The elocity of the ball after the impact The elocity of cube after the impact The elocity of cube after the impact 3 F t u The changing force that the spring get when the impulse happen The time caused during the impact The ratio of the energy dissipated to the total energy of the ball Comment: eery non-ector here is a scalar quantity. 3 Simplifying Assumptions When we analyze the models during the impact process in 4. and 4.3, we assume that the collision happens in the sweet spot. For the ball-cube model in 4. and 4.3, we assume a constant as the ratio of ball s dissipated energy to the ball s total energy to estimate the dissipation, considering the ball itself will dissipate quite a part of the energy obtained during the compression.

5 Team # 804 Page 5 of 3 Ignore the dissipation of the elastic potential energy stored in the bat. 4 The Model 4. Finding the sweet spot 4.. Illustration of the sweet spot There are a lot of definitions of sweet spot and een the so-called sweet zone. Howeer, the most common saying is that the sweet spot corresponds to the contact point that produces the least ibration in the batters grip. Here in this model, we focus on finding the zero ibration point on the bat by using seeral physical theorems and laws. 4.. A ball-bat collision model We analyze the process of the collision as follow: Three key time points are considered in the motion among the ball, the barrel and the hitter s hands. First: the ery moment the ball contacts the bat, i.e., the crack of the bat. Second: the moment the hitter s hands counteract the bat, i.e., the secondary ibration. Third: the moment the ball and the bat interact with each other after the secondary ibration. These time points discrete the whole process of hitting a baseball into four time interals as below: ball, First stage: before the ball hit the bat, the ball has an initial elocity while the bat has an initial elocity bat,. Second stage: between the crack of the bat and the secondary ibration. Third stage: after the secondary ibration and before the ball leaes the bat. Fourth stage: after the ball leaes the bat. According to the momentum theory and momentum conseration law, we arried at the following formulae associated with each impact: First impact: the ball impact with the bat while the total momentum of both is consered. Based on the momentum conseration law: r ball, + M r bat, = r ball, + M r bat, Second impact: impact between the bat and hands, the hitter can feel the ibration in the batter s r grip and gie the bat an impulse to resist the shaking force, which is recognized as I. According to the momentum theory: r r I = M bat,3 M r bat, Third impact: the ball and the bat interact with each other for a second time; the total momentum of both is consered. Based on the momentum conseration law:

6 Team # 804 Page 6 of 3 r ball,3 + M r bat,3 = r ball,4 + M r bat,4 r ball,3 = r ball, The whole process can be simplified like this: r r I = ( ball,4 + M r bat,4) ( r ball, + M r bat,) Figure : Illustration of the collision when the ball contacts with the bat. QuickTime?and a decompressor are needed to see this picture. r Suppose I has the amount I on the x-axis, the amount I x y on the y-axis. Since the amount of elocity of the bat on the x direction is zero, we get the motion formulae on both x and y directions. On the x axis I x = ball,4 sin β + M bat,4 ball, sinα M bat, On the y axis I y = ball,4 cosβ ball, cosα We now focus on the bat. Regard the handgrip point as the fulcrum O ; the z-axis is perpendicular to the paper sheet outward. The moment equation for rotation about z axis is expressed as: Jω = The moment of inertia J in the aboe formula can be calculated as: Accordingly, we arrie at: J = L z d dm

7 Team # 804 Page 7 of 3 ω bat,4 ω bat, = L z J Based on the conseration of angular momentum theory, we hae OC ball, + L bat, = OC ball,4 + L bat,4 Where L bat,i = Jω bat,i,i =, 4 We get ω bat,4 ω bat, = OC m( ball, ball,4 ) J According to theorem of motion of centre of mass, let the centre of mass B. The elocity of bat at the centre of mass: bat,i = OB ω bat,i,i =, 4 Simplify the motion formulae on both x and y directions with the aboe equations. I x = ball,4 sin β ball, sinα + M OB OC m( ball, ball,4 ) J Since C is the sweet spot, we want I x = 0 I y = 0 Get an equation ball,4 cosβ = ball, cosα Accordingly, the angle that the ball projects after hitting the bat should be as following in order to make the least forces transmitted to the hands: β = arccos( ball, cosα ) ball,4 The closer is β to a right angle, the larger ball,4 will be. When β is a right angle, the ball can gain the largest elocity to project outward. Howeer, β can neer be a right angle despite the case that α is also a right angle. As a conclusion, when α = β = 90 o The ball has a largest projecting speed. We can also come to the conclusion of the problem where the sweet spot should be: OC = J( ball,4 sin β ball, sinα) M OB( ball,4 ball, ) Examine the result to a real-world example

8 Team # 804 Page 8 of 3 The mass properties of a solid wood and a hollow metal bat are gien in the table. Bat Mass (g) C.G. (mm) MOI (kg m ) Ash Aluminum MOI: a bat s mass moment of inertia C.G.: the centre of graity In this case, we make the following assumptions to simplify the problem: The ball contacts the bat in a angle of 90; The ball leaes the bat with the highest speed, which means that the project angle is a right angle; The fulcrum, i.e. the grip point, is approximately to the end of the bat, and we suggest it is. Therefore, the sweet spot of the ash bat and the aluminum are 537.7mm and 785.5mm as we calculated. In sum, for a fixed fulcrum, the position of a sweet spot is associated with the mass of the bat, the mass moment of inertia and the centre of graity of the bat. Howeer, we need to recognize that each point in the handle is associated with a different sweet spot. For instance, professional players tend to hold the bat nearer to the end of the bat, which makes the sweet spot closer to the centre of graity. Analysis of the ibration when hitting out of the sweet spot For a gien spot C on the bat, the amount of x-axis ibration is I x = ball,4 sin β ball, sinα + M OB OC m( ball, ball,4 ) J We consider the case with the following assumptions: The ball contacts the bat in a angle of 90; The ball leaes the bat with the highest speed, which means that the project angle is a right angle; The fulcrum, i.e. the grip point, is fixed. Then we arrie at a simple notation: Notice that When I x = m( ball,4 ball, )( I x = 0 M OB OC J M OB OC ) J On the one hand, the hitting point C in the bat determines the ibration between the bat and hands when the mass of the bat, the mass moment of inertia and the centre of =

9 Team # 804 Page 9 of 3 graity of the bat are fixed. On the other hand, for a fixed hit point, the answer is arying based on the differences of M, OB and OC. We go on using the example of the ash bat and the aluminum bat in the aboe example. Fix the mass and elocity of the ball as below: Mass (kg) Initial elocity (m/s) Projecting elocity (m/s) Figure : illustration of how the x-axis impact aries with the change of the hitting point. From the figure, we can know that the wooden bat has a larger x-axis impact than a aluminum bat when the hit happen at the same distance from the sweet spot. In other words, the aluminum has a larger sweet spot zone. Simply put, according to the formula M OB OC I x = m( ball,4 ball, )( ) J The larger M or OB is, or the smaller J is, the more seere the impact is. 4. Corking

10 Team # 804 Page 0 of 3 In the following issue, we will establish two models to inestigate what consequences may cause when the bat changed. One model is about the mass of the bat; the other model is about the elasticity of the bat. 4.. Mass Effect Considering the bat and the ball. At the moment that the ball touches the bat, we can use the momentum conseration theory to sole the problem. Here we use the Coefficient of Restitution and get the conclusion by some references. Figure 3: Show the elocities of the ball and the bat in the collision process. V ball. V ball,4 V bat f Equations of the model From the aboe symbol explanation, we can conclude the following expressions: ball,4 + bat, f e = ball, m r = M As e A is the collision efficiency, so ball,4 = eaball, Since the impact between the bat and the ball is instantaneous, the effect of the collision force is great, so we can use the momentum conseration, and then we hae the following expression: bat, f = r( + ea) ball, We deduce the expressions as follows: ball,4 ea = rball, eball, eaball, = r ball,

11 Team # 804 Page of 3 e e = A r After coordinating the formulae, we get e r e = A r + Analysis Suppose that e=0.5, m=0.kg. Inestigate fie bats with different mass, M m r = M e r ea = + r The table lists some data about M and r, we calculate ea by the expression, then draw a figure as follows, which can help us analyze the problem. Figure 4: Here e is the coefficient of restitution, based on some datum we looked up from the references. Newton think that these two collision impulse is a constant ratio, and that the constant relates to the material of the collision objects, has nothing to do with the object shape, size and the speed before the collision. This constant is known as the Coefficient of Restitution, which is denoted by e.

12 Team # 804 Page of 3 There is another definition of the coefficient of restitution, namely, the Coefficient of Restitution is equal to the ratio of the absolute alue,which is the before and after relatie elocity of the collision of two objects. Based on the aboe conclusion and the explanation, we know that e is a constant, and has nothing to do with the elocity, so if we change the mass of the bat, e A arises changes. Results On the one hand, if the mass of the bat increases, based on the definition of r,we can know that r decreases, then using the aboe formula, we can conclude that e A decreases, then the collision efficiency decreases, the ball after the impact won t fly faster and farther. On the other hand, if the mass of the bat decreases, similarly to the analysis, we can know the collision efficiency increases, the ball after the impact will fly faster and farther, this increases the probability of homerun. Now we get the conclusion: lighter will be better! Therefore, corking a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) will decrease the mass of the bat. From this aspect, we can see corking a bat enhances the effect. Next, we will show the effect from the Coefficient of Restitution, i.e. from the material of the objects, the Coefficient of Restitution between the two objects is measured by experiment, and based on the reliable information, we can see that the corking bat causes the Coefficient of Restitution increases, compared to the former bat. From this aspect, we conclude that the corking bat enhances the effect, and increase the probability of homerun. 4.. Elasticity Effect Modeling Considering the wood bat, we can simplify the problem as the model in the Figure 5, here the string between the cube and cube simulate the property of the wood bat, since the wood bat is inflexible. The entirety of the cube, the string and cube simulate the wood bat. Then considering the corking bat, we can simplify the problem as the model in the Figure 6, here the spring between the cube and cube just like the stemming in the corking bat, that is, the stemming in the corking bat is simplified as the spring, since the stemming in the bat hae elasticity. The entirety of the cube, the spring and cube simulate the corking bat. Now we begin to analyze the two models as follows:

13 Team # 804 Page 3 of 3 Analysis The collision between a baseball and a bat is iolent, during the contact, the motion process of the ball will compress in a nonlinear way, comes to a momentary halt, reerses direction and then expands to its original shape. This process is inherently inefficient, with a large part of the ball's original kinetic energy dissipated in the internal structure of the ball. After analyzing the process of the impact, we can know that the initial elocities of the cube and cube is the same as well as the finally speed of the cube and cube is the same, while the speed of the cube and the speed of the cube is not equal in the process of the impact, which can be denoted by expressions as follow: At the beginning = 3 In the process 3 Finally = 3 3 Figure 5: The ball-cube model with a string QuickTime?and a decompressor are needed to see this picture. First, we consider the system which is composed of the ball and cube, which we call the partial system. After using the momentum conersation, an expression is concluded: m ( ) = 0 The left part of the expression represents the changes of the momentum. Then we hae the expression equal to the aboe: + = + Second, for the other system which is composed of the ball and cube and cube, here we call the whole system, we employ the conersation of energy theory : + ( m+ m3) = Etotal + ( m+ m3) Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not perfectly elastic collision, thus by the ratio u, we can get the following expression in which E total represent the total energy the ball get after the collision = ( u) E total Then the expression equals to the next:

14 Team # 804 Page 4 of 3 Etotal = u So we hae the following expression: + ( m+ m3) = + ( m+ m3) u The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system. Figure 6: The ball-cube model with a spring QuickTime?and a decompressor are needed to see this picture. First, we consider the system which is composed of the ball and cube, which we call the partial system, we use the momentum theory, then an expression is concluded: F t = m ( m ) The right part of the expression represents the changes of the momentum, and F t represents the impulse during the impact. Then we hae the expression equal to the aboe one: + = F t+ + Second, for the other system which is composed of the ball and cube and cube, which we call the whole system, the total energy after the impact in the whole system equals to the total energy before the impact. We employ the conersation of energy theory: + ( m+ m3) = Etotal + ( m+ m3) Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not perfectly elastic collision. Thus, by the ratio u, we can get the following expressions in which E total represent the total energy the ball get after the collision. = ( u) E total Then the expression equals to the next: Etotal = u So we hae the following expression: + ( m+ m3) = + ( m+ m3) u The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system.

15 Team # 804 Page 5 of 3 Results From the expressions listed aboe, we can calculate in the second model is larger than in the first model, then return to the real-world problem, we can get the conclusion that the speed of the ball after hitting the corking bat is larger than the wood bat, so the corking bat makes it easier for player to achiee better results. Whereas, as we all know, the ductility of the corking bat may be a little better than the wood, so when the bat is used in a competition, a wood bat is more easily to be broken off, this of course influences the result. As the first issue which we come up with the sweet spot, we can conclude by some important information that the optimal hitting area changes with the corking bat, and the optimal hitting area of the metal bat may be larger than the wood bat, which makes it easier when the player hits. Therefore, our sportsmanship is to improe the baseball player's own physical fitness and skills, rather than engage in sports equipment competitions,regular competitions is not allowed to use the corking bat. 4.3 Wood or Metal In this issue, we will also establish two models, one model is about the mass, and the other model is about the elasticity Mass Model It is common sense that a aluminum bat is far more lighter than a wooden bat in weight. The model considering the mass here is the same as the first model in the corking bat, so we won t gie unnecessary details Elasticity Model Comment: similarly to the former model but not the same Considering the wood bat, we can simplify the problem as the model in the Figure 7, here the string between the cube and cube simulate the property of the wood bat, since the wood bat is inflexible. The entirety of the cube, the string and cube simulate the wood bat. Then considering the metal bat, we can simplify the problem as the model in the Figure 8, here the spring between the cube and cube just like the metal bat, howeer, the differences is cube represent the surface of the metal bat. To make the formerly model also fit this problem, we can get m much smaller, then the problem can also be soled by the formerly model. Whereas, the mechanism is totally different: for the corking bat, the elasticity comes from the stemming inside the corking bat. While for the metal bat, the elasticity comes from the flexible surface of the metal, which arises the trampoline effect. Then we hae a result that although the models of the two problems are similar, they hae different mechanism.

16 Team # 804 Page 6 of 3 Now we begin to analyze the two models as follows: Analysis The collision between a baseball and a bat is iolent. During the contact, the motion process of the ball will compress in a nonlinear way, comes to a momentary halt, reerses direction and then expands to its original shape. This process is inherently inefficient, with a large part of the ball's original kinetic energy dissipated in the internal structure of the ball. For the collision of a baseball with a bat, the ball and the bat mutually compress each other during the collision, so that some of the ball s total energy that might otherwise hae gone into compressing the ball goes into compressing the bat instead. Therefore less energy gets stored and dissipated in the ball. To compare the hit effect of the solid wood bat and the corked wood bat, we should highlight the point that the ball s after-impact speed is much more largely depends on how effectiely the compressed energy stored in the bat and then returned to the ball. When a ball hits a solid wood bat, the bat barrel hardly compress, while the ball compresses to nearly half its original diameter, losing up to 75% of its initial energy to internal friction forces. Howeer, in a hollow bat, such as an aluminum bat, the bat barrel compresses somewhat like a spring. This means that the ball is not compressed as much and loses less energy to internal friction forces. Furthermore, in a metal hollow bat, most of the energy temporarily stored in the compressed bat is returned to the ball, which results in the ball's after-impact speed much higher, compared with the ball's colliding with a solid wood bat. This phenomenon is also known as the trampoline effect. After analyzing the process of the impact, we know that the initial elocities of the cube and cube are the same as well as the final elocities of the cube and cube, while the speed of the cube and the speed of the cube is not equal in the process of the impact, which can be denoted by expressions as follows: At the beginning = 3 In the process 3 Finally = 3 3 Figure 7: Ball-cube model with a string QuickTime?and a decompressor are needed to see this picture.

17 Team # 804 Page 7 of 3 First, we consider the system which is composed of the ball and cube,here we call the partial system, we use the momentum conersation, then an expression is concluded: m ( ) =0 The left part of the expression represents the changes of the momentum. Then we hae the expression equal to the aboe one: + = + Second, for the other system which is composed of the ball and cube and cube, here we call the whole system, we employ the conersation of energy theory : + ( m+ m3) = Etotal + ( m+ m3) Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not Perfectly elastic collision, thus, by the ratio u, we can get the following expression, in which E total represent the total energy the ball get after the collision = ( u) E total Then the expression equals to the next: Etotal = u So we hae the following expression: + ( m+ m3) = + ( m+ m3) u The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system. Figure 8: Ball-cube model with a spring. QuickTime?and a decompressor are needed to see this picture. First, we consider the system which is composed of the ball and cube,here we call the partial system, we use the momentum theory, then an expression is concluded: F t= m ( ) The right part of the expression represents the changes of the momentum, and F t represents the impulse during the impact. Then we hae the expression equal to the aboe one: + = F t+ + Second, for the other system which is composed of the ball and cube and cube, here we call the whole system, the total energy after the impact in the whole system

18 Team # 804 Page 8 of 3 equals to the total energy before the impact. We employ the conersation of energy theory: + ( m+ m3) = Etotal + ( m+ m3) Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not Perfectly elastic collision, thus, by the ratio u, we can get the following expression in which E total represent the total energy the ball get after the collision: = ( u) E total Then the expression equals to the next: Etotal = u So we hae the following expression: + ( m+ m3) = + ( m+ m3) u The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system. Results From the expressions listed aboe, we can calculate that in the second model is larger than in the first model. Considering the real-world problem, we can get the conclusion that the speed of the ball after hitting at a metal bat is larger than that at a wood bat, so the metal bat makes it easier for players to achiee better results Other factors concerned Using metal bats is dangerous in some ways, especially in formal games. Firstly, the repeated friction between metal bat and the player's skin will produce a large number of static electricity, which we can know from electromagnetic theorems. Then the static charge will attract dust in the air, water droplets, etc. thus makes the bat slipping. Secondly, a metal bat can gie the ball so greater a speed that een beyond people s ability to react to hit or aoid that object, this may also cause some dangerous situations like hurt the other players or een the audiences. We all know that the ductility of metal is much better than the wood, so when the bat is used in a competition, a wood bat is more easily to be broken off, which may affect the procedure of the competition. And for the good ductility, the metal bat is more flexible, which helps the player to make good achieement. Howeer, the metal bat are prohibited in competitions for it is unsafe and unfair, the most significant

19 Team # 804 Page 9 of 3 maters to considered in eery legal competition. In contrast, for amateurs, metal bats are ery well welcomed for its low weight, good ductility and easy to handle. Compare with a wooden bat, a metal bat can also help to decrease the ibration of the bat to a lower scale for it is much lighter than a wooden bat. A lighter bat can help to reduce the ibration when hit outside of the sweet spot in spite of the fact that whateer a hit is on the sweet spot the ibration is zero in the x-axis, as we hae discussed in the first model. This phenomenon gies the players a lot of help in the competition, not only in physical ways but also in psychology ways. And if one uses the metal bat, he or she may hae more time to identify the ball, and we can hit the ball faster. From the first model where we come up with the sweet spot, we can conclude important information that the optimal hitting area changes with the material of the bat, and the optimal hitting area of the metal bat is larger than the wood bat, which makes it easier when the player hits. In sum, our sportsmanship is to improe the baseball player's own physical fitness and skills, rather than engage sports equipment competitions. Therefore, regular competitions are not allowed to use metal bat. 5 Comparison between Corking and metal bat Considering the important factors, the differences between the corking bat and the metal bat are as follows: The metal bat can arise the trampoline effect, which the corking bat cannot, that makes the ball projects faster and farther. Neertheless, the corking bat has the stemming that is flexible, while the metal bat don t. So the corking bat and the metal bat both can make the player gain good scores. Of course, the corking bat and the metal bat are both lighter than the wood bat, which are more adantageous than the wood bat. There are also some other factors that make the corking bat and the metal bat better, which can be found in the paper. 6 Strengths and Weaknesses Weaknesses: For the trampoline effect, we do not gie a logical and strict proof and experiment to proe the trampoline effect. Instead, we analyze the trampoline effect and its function in the model. The next we should do is to do experiments to strictly analyze the trampoline effect. When mentioning additional reasons to explain the problems, we gie statements. This aspect can be improed. The coefficient of restitution should be inestigated experimentally, but being limited to the condition, we use the scholars conclusion. Strengths:

20 Team # 804 Page 0 of 3 In the second problem and the third problem, we simplify the model that is conincing and accustomed to the issue. I think this model is an excellent part of the paper. What s more, in the first model of the second problem, it s a good idea using ea to sole the problem. It s an innoation in the paper, because we use ea instead of the ball s speed to sole the problem. In the second model of the second problem, we simplify the corking bat as two cubes and a spring. We transform the question into a physical analysis. Thus we can sole the issue. While in the second model of the third problem, we also simplify the issue as two cubes and a spring. But the mechanism is quite different. 7 Further discussions For the trampoline effect, we may make further discussion. Since the wood is not flexible and is solid, the wood bat won t hae the trampoline effect. This is the difference between the wood bat (or the corking bat) and the metal bat. Howeer, we can do experiments to proe the trampoline effect. Howeer, the trampoline effect makes the ball get more efficient energy. Just because of the elasticity of the surface of the metal, the metal bat holds a part of the energy so that the energy dissipates less. It s the mechanism of the trampoline effect. Its existence is because the ball and the bat dissipate energy itself. Since we haen t consider the dissipate energy in the bat. Here we can make a further discussion. First we should assume that u is a constant. Here u is the ratio of the energy dissipated to the total energy of the bat. Then we just adopt the model of the third problem. The model and the symbols are also the same. Here we can just begin from the equations. Equations of figure 5 First, we consider the system which is composed of the ball and cube, here we call the partial system, we use the momentum conersation, then an expression is concluded: m ( ) = 0 The left part of the expression represents the changes of the momentum. Then we hae the expression equal to the aboe one: + = + Second, when dealing with the other system that is composed of the ball and cube and cube, here we call the whole system, we employ the conersation of energy theory: + ( m+ m3 ) = Etotal + E total Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not Perfectly elastic collision, thus, by the ratio u, we can get the following expression in which E total represent the total energy the ball get after the collision

21 Team # 804 Page of 3 = ( u) E total Then the expression equals to the next: Etotal = u Similarly we hae the expression ( m + m3 ) = ( u ) E total So we hae the following expression: ( 3) ( 3) + m + m = u + u m + m The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system. Equations of Figure 6 First, we consider the system which is composed of the ball and cube, here we call the partial system, we use the momentum conersation, then an expression is concluded: F t= m ( ) The left part of the expression represents the changes of the momentum. Then we hae the expression equal to the aboe one: + = F t+ + Second, for the other system, which is composed of the ball and cube and cube, here we call the whole system, we employ the conersation of energy theory: + ( m+ m3 ) = Etotal + E total Since we consider the dissipated energy of the ball in the impact between the ball and the bat, that is, the collision between the ball and the bat is not Perfectly elastic collision, thus, by the ratio u, we can get the following expressions in which E total represent the total energy the ball get after the collision = ( u) E total Then the expression equals to the next: Etotal = u Similarly we hae the expression ( m + m3 ) = ( u ) E total So we hae the following expression: ( 3) ( 3) + m + m = u + u m + m The left part denotes the energy before the impact in the whole system, and the right part denotes the energy after the impact in the whole system. Results

22 Team # 804 Page of 3 From the expressions listed aboe, we can calculate than in the second model is larger in the first model, then return to the real-world problem, we can get the conclusion that the speed of the ball after hitting by the metal bat is larger than the wood bat. As a conclusion, whether we consider the dissipated energy or not, the result is the same. Therefore the simplified model is correct and supports our opinion toughly. 8 References [] [] Mechanical properties of materials/editor: Chunting Liu, Ji Ma

23 Team # 804 Page 3 of 3 [3] Sociodymamics A Systemic Approach To Mathematical Modelling In The Social Sciences [4] K.Mechanical properties of materials/yu Haisheng, Sergiy N, Shukaye,Manoun Medraj. [5] [6] [7] Mechanics of materials/r.c.hibbeler.5 th ed. [8] [9]