Introduction
Bowling is intrinsic to the game of cricket, and it can be said that it is half of the game. One way of getting the batsman out in cricket is by hitting the stumps on a ball, by getting it past the batsman’s defence. Similarly, hitting the batsman’s body in front of the stumps, where it can be deemed that, without the batsman there, the ball would have hit the stumps, will result in the batsman being given out by the umpire, this is referred to as a leg before wicket, or an LBW.
Information about the trajectory of the pitching ball is vital to almost all individuals on a cricket field. For the bowler, whether they be fast or spin bowlers, they should try to pitch the ball in such a way that it will be in line with the stumps, even after pitching. For the batsman, predicting the trajectory of the ball, after pitching is extremely vital to their defence and offence, as it will determine the shots they will play. If the batsman deems that his stumps are in danger, he will usually use a defensive shot against the pitch, whereas a pitch missing the stumps will usually be attacked. For the umpire, predicting the trajectory of the ball is extremely important, as the umpire makes the judgement of whether the ball would have hit the stumps or not.
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Projectile motion describes motion that is experienced by a mass or “projectile” thrown on earth, which moves along a path, solely under the action of gravity. Air resistance is considered negligible in this motion. As this investigation focuses on a ball after it bounces, it is still considered to be a projectile and follows the rules of projectile motion.
Rationale
This investigation will focus on finding the trajectory of a cricket ball after the ball has bounced, and will create models for this trajectory, with and without drag force. This will then be compared to a real life ball bowled by me. I have used a camera to film myself bowling a ball, and it’s trajectory can be calculated using a motion logger. I will be using the motion logger “Logger Pro” by Vernier in order to conduct my investigation. In my investigation, I will also see how the bowler can hit the stumps, and whether my model will go on to hit the stumps like the ball I myself have bowled. I will first investigate the projectile motion of the ball with only gravity acting upon it. I will then add in the drag force experienced by the ball. Last of all, I will attempt to compare the trajectories, theoretical and experimental.
I have chosen this investigation as I am a passionate player of cricket, and hence am interested in how the balls I bowl can actually be modelled using physics and mathematics, and how the theoretical trajectory of the ball can be traced, even after the ball has hit the batsman.
The situation that I am looking at can be described as below
The ball starts at x0, which is any point on the Cartesian plane, and the stumps are at xs which is a point to the right of x0 on the Cartesian plane. In this investigation, the stumps are kept 3 metres from the origin, as this is their position from the bounce when I filmed my own ball being bowled. Hence x0 = 0, and xs = 3 The stumps are 0.71 metres tall, as stated by the official ICC regulations. This investigation assumes that the ball has already bounced, and the projectile motion of the ball will be calculated from after the pitch of the ball. As the ball is only allowed to bounce one, hence this investigation will not look at the ball bouncing on the pitch again. The parabola of the ball will hence either be hitting the stumps or going over them.
As a cricket ball, or any projectile, for that matter, flies through the air, it is imperative that due to ambient conditions, there will be resistance by the air against the ball. This is because of the fact that air pressure on the front of the projectile will be greater than the force applied to the back of the projectile. This force is called the drag force. Drag force is obviously impacted by many different factors, including the speed of the ball, the size of the ball, surface area of the ball and the density of the air around the ball.
This shows that as the area of the cricket ball or any projectile, increases, the drag force will also increase, whilst everything else is constant, as there will be more resistance against the ball due to the larger area. Furthermore drag will also increase as air density increases, as there are more particles of air in the atmosphere, increasing the resistance against the ball.
It is clear from this graph, that there is a great discrepancy in the graphs of theoretical and actual ball trajectories. This is likely due to the fact that there are many uncertainties associated with these calculations, especially as Magnus force and other factors have not been taken into account. However, it is clear that whatever the trajectory of the balls, all of them would have gone on to hit the stumps, as can be seen below. Both lines y=0.71, and x=3, intersect with the trajectories of the balls.
Whilst there are definitely discrepancies between the theoretical model and the actual experimental trajectory, by slightly changing some variables, for example the angle at x=0, velocity, and drag & lift coefficient, the trajectory can be seen better.
I tried to fix the variables to see if my trajectory could be mapped with greater accuracy. By increasing the angle slightly, up to 15 degrees instead of 12, and increasing the drag coefficient up to 0.7, the drag trajectory (black) can be seen to be nearly perfectly in line with my actual trajectory (blue). This also results in the hitting of the stumps at just below the 0.71 mark, in the theoretical trajectory.
As these variables are very difficult to calculate accurately, it is very likely that the parameters used were not the exact same as their actual values, and these values could be closer to the ones used in the second graph.
It is clear that very minute differences in variables can result in great changes in cricket. A small variation in the angle of a pitch, or some variations in the air density could be the difference between getting a wicket, and getting your ball smashed for a 6. It is hence clear that, whilst practice and skill are on one hand a great factor in the results a cricket player produces, the player can only, to an extent, control the pitching and speed of the ball, the rest definitely depends on other factors, some of which have been explored here.
Conclusion
This investigation explored the projectile motion of a cricket ball after it was bowled. Two different models for the motion of the ball were created, one taking into account drag force, whilst the other did not. Three different curves were graphed using these models, parameters for which were obtained from an experimental trajectory of a ball which was filmed. A comparison of these different curves was conducted and it was determined that the curves of the theoretical trajectories were wider than the curve of the actual trajectory, and resulted in a greater range, hence there was a discrepancy between the theoretical and actual curves, although the discrepancy was lower for the curve which took drag into account. Finally, it was determined that slightly changing some variables resulted in a better indication of the trajectory of the ball, thus showing how even minute changes resulted in great differences in the models.
Taking into account the assumptions made during this investigation, and the limitations faced, this investigation has successfully fulfilled its aim of modelling the trajectory of a cricket ball.
Assumptions and limitations
The reliability of this investigation, and the models produced through the investigation, may be hindered by the fact that there are many limitations and assumption made whilst investigating this subject.
First of all, it is assumed that there are only drag and gravity acting upon the ball. This is a huge assumption, as there are many other forces acting upon the ball, and not just the two explored, for example, Magnus force, which is the lift of the ball. This is most likely the greatest assumption made, and much of the discrepancy between the theoretical and actual trajectories is very likely due to this assumption. Had the other forces been taken into account, it is safe to assume that such a large discrepancy would not have occurred. However, the addition of these other forces, whilst initially considered, was deemed extremely complex, and was not in the scope of the curriculum.
Another assumption that was made was that the drag coefficient was deemed to be constant throughout the motion of the ball. However, it is important to note that this coefficient would have decreased, as drag force decreases the spin of the ball, and hence decreases the coefficient. Furthermore, the condition of the ball was assumed to be in agreement with the standards of the ICC. IN reality, this would definitely not have been the case, as when a cricket ball is used, it changes shape, and loses mass. This slight variation would also have caused discrepancies in the trajectory calculated and the actual trajectory of the ball.
Furthermore, it was assumed that the ball was moving completely straight when mapping the ball, and any sideways movement was ignored. In reality, it is obvious that it would not have been moving in a strictly forwards motion, and there would have been sideways motion of the ball.
Whilst there are many limitations limiting the reliability of this report, it would be possible to increase the scope of this investigation, hence reducing the uncertainty and error. This would also result in a better calculation of the trajectory of the cricket ball. For example, instead of looking at just the 2 dimensional motion, the 3 dimensional motion could be investigated. Furthermore, the exact values of the variables can be calculated. This could prove to be a valuable tool for players and coaches alike, in order to improve the players accuracy whilst bowling. It could also prove valuable as an alternative to hawkeye, a current trajectory calculating program used to calculate whether a batsman is LBW or not. Hawk-eye uses 3d mapping and statistics along with artificial intelligence to calculate the trajectory, however, it is very controversial, and a more accurate mathematical model could very much replace hawk-eye in the future.
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