Throughout high school, track and field has been one of the sports that I have competed in, in particular the javelin throw. With my coach also saying I should release the javelin with more power/speed and at a greater angle of release, I have always wondered whether changing the speed and angle of the the javelin at the point of release would actually change the outcome of the distance that would be thrown  The javelin throw requires a great amount of technique and flexibility in the speed and angle that the javelin is thrown at in order to produce a greater distance. When I threw the javelin during practice I noticed that the angle I would throw the javelin was greatly affected by the speed I threw it at. I also noticed that due to a change a speed there would be a change of the angle that the javelin was thrown at which would also affect the distance thrown and thus the parabolic trajectory of the throw. I have also watched top-level javelin throwers during competition and noticed that the throwers tended to increase their speed greatly with a run up before the release for the throw which results in the release of the javelin to be at a high angle and this is normally seen as the optimum throw technique for professional competitions in the javelin throw. Thus I wanted to show my mathematical knowledge in kinematics and trigonometry into one of my main sports. The ideal women’s javelin throw is similar to a parabolic curve, which means that the speed at which you throw the javelin is crucial as it also affects the angle of the throw.  The ideal throw is needed in order for the distance thrown to be increased. However, with people and women having different heights from where they throw the javelin based on how tall they are. This requires different individuals to adjust their speed in order to produce the ideal angle of release for the ideal javelin throw. As a short individual, I often compete in competitions and find that because of my height, I have to adjust my speed to extend the range I can throw the javelin. Hence, the aim of this investigation is to explore how the speed and thus angle at which the individual throws the javelin influences the horizontal range. One hundred percent of the time, individuals have to throw a range that exceeds other people and thus requires an optimum angle and speed for this to occur. Furthermore, the investigation also aims to model the ideal throw that results in the largest horizontal range with a quadratic equation.   Gathering DataTo gather the data needed, I had to film a video of myself throwing a javelin at different angles and speeds. I changed the speed and angle that I threw the javelin at for every experimental trial in order to vary the two variables. I managed to film a total of 20 experimental throws with 2 being at each angle of 22.58o, 32.25o, 39.40o, 47.90o, 59.70o and varying the speed. After conducting research and reviewing the videos of the javelin throw, I decided that the angle that resulted in the clearest and best model of a parabolic curve with a large horizontal range was the angle that resulted in the 12m vertical maximum height at 39.40o. Therefore for the modelling of the quadratic equation, the height of 12m was kept constant for the investigation. When filming the throws the camera was placed on a tripod.  Past research and filmings of the javelin throw showed that the point of release and thus angle of attack of the javelin is the most precise when looking at the back end of the javelin just before release. Thus, I decided to use the bottom and lower point of the javelin as the focal point from where I measured the angle from throughout the experiment. To keep the investigation controlled the same standard women’s javelin was used for all trials and throws.  Finding the speed and angle of the throwAfter filming the throw’s, I was able to upload and import the clips of the throws being thrown at different angles and speeds onto a graphing program and software called Tracker. While filming I had placed a marker to show where the point of release would be from which was 5m from the start of the run. When looking at the video, I could then use Tracker to find out the time it took for the run up for the throw by using the equation: Speed (ms-1)=Distance (m)Time (s)Example 1: For Trial 1Distance = 11.70mTime = 2sSpeed=11.702Speed=5.85ms-1 To enable the measurement of the angle that the javelin was released at, screenshots from the video were than uploaded on a graph on the graphing program used called DESMOS, with the photo being positioned with the lower point of the javelin starting at the coordinates (0,0). In order to find the angle of release and attack of the javelin, a line with a linear gradient was placed tracking the path of the javelin to its highest point and is shown in Figure 1 in Trial 1 seen below, where the green line follows the line of the javelin, the javelins highest point is where the orange line and green intersect. The picture below was one of the 10 screenshots from the video. Figure 1: Initial angle of release and method used to find the angleFrom that, the coordinates of the highest point that the javelin reached were found on the x and y-axis and provided the lengths of two sides of a triangle. The angle of the throw was than easy to find by using the inverse tangent. From previously obtained informationtan ()=OppositeAdjacent so tan-1(OppositeAdjacent) = Example 2: For Trial 1tan ()=2.445.85 so tan-1(2.445.85) =This method was used to find the angle for all experimental trials. Finding the ideal equation for the horizontal range of projectile motion for the javelin throwProjectile motion incorporates two separate entities, vertical distance and horizontal distance. The horizontal range is one that is based entirely off of the horizontal distance of the model. When changing the speed at which I threw the javelin I noticed a change in the angles that it was being released from and thus decided to control the angles at which I threw at, that ranged from 23° to 60°. I did this in order to see how the change in both speed and angle simultaneously work together to change the horizontal range of the javelin throw. The effect of angle is seen below in Figure 1 for an object in another math problem. Figure 1: Effect of angle on an object’s horizontal distance. Image from: LumenLearning.comIn order to find the horizontal range of projectile motion referred to as R it is important for one to take into account the angle of attack (), initial velocity (V) and the acceleration in regards to gravity where g=9.8 ms-1. With these variables you can use the equation below to find the horizontal range of projectile motion:First equation: R=(V)2sin(2)gIn order to find the maximum height of the javelin in projectile motion the following equation uses the same variables with H equalling the maximum height:Second equation: H=(V)2sin()22g However, when trying to find both the maximum height of the javelin and the horizontal range, I found that I had no value for the initial velocity of the object and thus had to decide that the maximum height reached by the javelin would be the one that resulted in the largest horizontal range during my experiment which was Trial 3 with an angle of 39.4° and had a maximum height of 12m. With the variable of the maximum being set to 12m, I was then able to rearrange the second equation to isolate the initial velocity, V, to create the third equation, as seen below: H=(V)2sin()22g(2g) (H)=(V)2sin()2(2g) (H)sin()2=(V)2Third equation: (2g) (H)sin()2=VNow it was possible for me to find the horizontal range for any angle, with first finding the initial velocity through the third equation and plugging it into the first equation, creating the final and fourth equation:Final equation: R=(2g) (H)sin()2sin(2)gExperimental ResultsUsing the final equation above to find the horizontal range the data below was acquired and graphed on a scatter plot to show the relationship between the horizontal range and the angle of the throw that was initially affected by the speed. Table 1: The Maximum Height, Approachment speed, angle, initial velocity and horizontal range of 5 experimental trialsTrialMax Height (m)Approachment speed (s)Angle of Throw (°)Initial Velocity (m/s)Horizontal Range (m)112.01.122.623.138.5212.01.232.323.550.7312.01.739.426.059.9412.01.447.923.957.8512.01.559.724.258.4Graph 1: Then Angle of Release of Javelin Throw (°) versus the horizontal distance (m) reached with a maximum throw height of 12mThe calculations that helped produce the above graph and data suggest that with a constant maximum throw height of 12m, the distance covered by a javelin increases as the angle of the release is increased until around 40°. If the angle goes beyond 40°, the javelin will start to drop earlier due to gravity and results in the lower horizontal distance. The data also shows that as the speed increases the angle of release increases and thus results in the above trend between angle of release and horizontal range. Therefore, there is a direct influence of speed and angle on the horizontal distance of a javelin throw. In general, the correlation and relationship between the angle and speed of release of the javelin and the horizontal range is a positive relationship to a threshold angle of approximately 40°, where the relationship starts to become regative. In order to provide a clearer representation of the javelin parabolic throw, I decided to model the trial 3 throw as it gave me the longest horizontal distance. The modeling of this throw was to model the optimal throw for the women’s javelin throw. To model the largest horizontal distance I plotted the maximum height (12, 0), point of throw (-29.95, 0), point of landing (29.95, 0)  as well as two points halfway between the maximum heights and the respective point of release and landing on Desmos with the help of the screenshots taken previously. I managed to derive the graph below with the previously mentioned points:Graph 2: Plotted points from modeling of throwHaving the maximum, x and y intercepts, it was possible for me to create a parabolic function that modelled a throw that resulted in the longest distance. The first quadratic equation I started with was in vertex form and included information that is previously known:Vertex Form Equation: y=a(x-h)2+kWith the maximum point being (0,12), k = 12. Hence: y=a(x-0)2+12In order to find ‘a’, the us of a point on the graph was used, in this case I decided to use the x intercept as the point to find ‘a’0=a(-29.95-0)2+120=897a+12-12897=a-0.013=aAfter finding ‘a’ I was able to complete the model the throw with the quadratic equation produced with x being the horizontal distance traveled and y being the height: FINAL EQUATION: y=-0.013×2+12Domain: {–29.95 ? x ? 29.95}Range: {0 ? y ? 12}Graph 3: Ideal Parabolic Curve of javelin throwThe graphs range being between 0 and 12 is being the maximum height of the throw was chosen to be 12m, therefore the x axis and line y=0 represents ground level with the maximum height being y=12. In order to model all the throws on the same coordinate grid, the same method used above was implemented to show the following parabolas for the experimental trials. The x axis below is the horizontal range (m) and the y axis the the height of the throw (m).Graph 4: Parabolic curves of the 5 experimental throws:Table 2: Parabolic Curves Data of the 5 experimental throws:LegendTrial NumberAngle of Throw (°)Horizontal Range (m)122.638.5232.350.7339.459.9447.957.8559.758.4The parabolas provided from the different experimental throws provided information about the relationship between angle and horizontal range. It is noticed that as the angle of release neared 40o, the horizontal area and range covered by parabola was widest and had a larger vertical dilation factor. For example, the orange parabola represents a throw with an angle of release of 22.6o and the smallest horizontal range of 38.5m, around half the angle that provided the largest horizontal range. Secondly it is seen that after angles pass the threshold angle of 40o there starts to be a slight decrease in the horizontal range that the javelin can be thrown at as seen in trials 4 and 5 having smaller horizontal ranges than that of trial 3.   It was noticed that as there is an increase in speed the angle tends to change with a large speed being able to produce the optimum angle of approximately 40o. This characteristic suggests that when a javelin thrower increases their speed during their run up towards the point of release, it is more likely that they will throw the javelin at the optimum angle needed to produce a larger horizontal distance and range. Hence, it is ideal for professional javelin throwers to have a fast speed when approaching the release of the javelin. If the speed is too slow, the throw is more likely going to have an angle that is either too small resulting in the javelin falling to the ground earlier or the angle will become too large and result in a dipping of the javelin too early which in turn results in a smaller horizontal distance and range. In relation to my personal competition of the javelin throw, I realised the reason as to why my coach asks me to try and perfect a fast run-up towards the point of releasing the javelin, rather than have a slow throw from a standstill or slow run-up. Thus, in the javelin throw the most desirable way to produce a large horizontal distance, the individual must have a fast speed in order to release the javelin at the optimum angle for a large horizontal distance. Conclusion In conclusion, speeds effect from a run up on the angle of release of the javelin plays a sufficient and large role in the final horizontal range covered by the javelin after release. Overall, as the speed of the javelin run-up increases the angle at which the javelin is thrown at is more likely to be at the optimal angle of 40o that produces a greater horizontal distance. Thus, the data suggests that if a javelin thrower is wanting to create a larger horizontal distance, it is ideal to approach the release with a fast run-up as it ensures that the angle of release will close to that of the threshold and optimal angle needed to produce the large horizontal range. When the angle of the javelin is below or exceeds that of the optimal angle the horizontal range decreases and the parabolic curve becomes narrower, but with an increase of speed the javelins’ angle of release reaches that of the optimal angle and results in a wider parabolic curve meaning a larger horizontal range. However, the ideal speed at which a javelin thrower should approach a thrower would need to be found through further reconstruction of the equations in the investigation and research. Overall, in my competition in javelin, I have found that the best way to increase my distances of my throws is to ensure that I have a fast run-up that allows me to release the javelin at an optimal angle and to focus on the angle I throw it at being close to 40o. EvaluationOverall, the experiment of the effect of speed and thus angle on the horizontal distance covered by a women’s javelin was able to be investigated thoroughly due to a sufficient amount of data collected to confirm the trend and data. However, there were a small number of limitations that could have resulted on an impact of the final data outcomes and processing. Initially, within the investigation it is assumed that the distance travelled from the point of the release to the final horizontal was on the same level, when the point of release is higher than the level that the javelin lands. This is because the release of the javelin is from an individual’s hand and lands at ground level. This could have effect the horizontal range that was the outcome as it affects the height from which it was released and thus also effects the maximum height. In the future, the best way to make sure that the horizontal distance is not affected by this is to take the distance that the javelin has traveled when it is at the height of the individuals hand when the javelin is released. A second limitation that I had was having to find the initial velocity of the javelin throw through the equations. I managed to overcome this by using the maximum height formula, but this resulted in having to use twelve meters as a constant maximum height when in reality it was most likely slightly higher or lower than the controlled twelve meters. However, this allowed me to solve the equation to find the ideal javelin throw angle which led me to the modeling of the optimal javelin throw. Finally, graphing a parabolic curve of the throws on an online software could have resulted in the parabolic curve being slightly off and not accurately representing each trials parabolic curve path that occured in reality. When solving the angle the position of the line following the javelin was an estimation, thus the angles that the javelins were released at in this investigation may not be precise enough and hence impacts the horizontal range of the javelin through. Ideally, the best way to ensure that the angle is as precise as possible is to use a software the tracks the javelin in a video and gives out a precise angle. Overall, I feel that the investigation allowed for me to overcome many of these challenges and limitations by finding new ways to solve mathematical problems that arise in everyday life and also made me see the usefulness of knowing what relationships different variables have such as the one between speed, angle and horizontal range of a javelin throw.