To find out how the starting position of a ski jumper affects the horizontal distance travelled in the jump.
We will model the jumper and the ski slope as a ball bearing and a curtain rail. We will not take air resistance, friction and other various type of energy lost into account. However in practical we have to keep in mind that they do exist and causes variation in our results.Factors which affects the range:Changing the vertical height of drop from the slope (‘h’) would vary the range. This is because according to the law of conservation of energy, it cannot be made or destroyed but transferred. To apply this law into this practical, we can say all of the GPE (Gravitational Potential Energy) is converted into KE (Kinetic Energy) assuming no energy is wasted. As a result, GPE is equal to KE. Hence if mass (kg) and acceleration (ms-2) due to gravity remains constant, varying the height (metres) would directly affect the velocity (ms-1).
mgh = 1/2 mv2So we are saying GPE lost causes a gain in KE. Ultimately, the greater the height dropped the greater the velocity it has when leaving the ramp and consequently greater the range. With this theory, we can confidently say that changing the gradient would not make a difference to the range if the height remains constant.
What will happen is that the object simply accelerates faster but the time varies. However if the drop height on slope and other factors remains the same, the velocity of the ball leaving the ramp should be theoretically, identical. Furthermore it is a very difficult task to perform, as the ramp needs to bend and unbend, thus deteriorating the accuracy.The launch angle is another factor affecting the range.
This is because it gives the particle a vertical velocity component rather than just a horizontal component. As a result this varies the time period of the object in the air. However in practical, it is difficult to alter the launch angle as the equipments are fixed. It would cause significant errors and therefore it would not be a good variable for this investigation.The drop height(‘s’) is an obvious variable as it varies the time of flight in the air thus changing the range. But it is a difficult one to adjust as we don’t have any appropriate apparatus to do this accurately and systematically. Hence this is not a suitable variable to investigate.HypothesisWe have decided to investigate how the vertical height of drop on the slope relates with the range achieved.
As explained earlier on, the greater the height, the greater the range. This is because more GPE [mgh = mass (kg) * acceleration due to gravity (ms-2) * height of drop (m)] is converted into KE [1/2mv2 = 1/2 * mass (kg) * velocity (ms-1)]. Since the two masses will cancel out while acceleration due to gravity is assumed to remain constant, we can say that as height increases, the velocity gain would increase proportionally. By neglecting the existence of friction, air drag and other form of resistance causing energy being wasted, we are assuming that all GPE is transferred into KE.
Theoretically I am correct, but in real life all these complications exist and I might find varied results.To calculate the range, we have to find the horizontal and vertical component separately. Since we are ignoring any type of resistance, the ball which is modelled as a particle should have a constant horizontal velocity at the moment it reaches zero GPE level. This is when the ball is at the point of the ramp where it is parallel with the table. This is the point of maximum KE and zero GPE, hence the ball should be at a constant speed. At the moment it leaves the ramp it will move freely under gravity causing only a downward acceleration. As a result the drop height has no effect on the horizontal velocity but vertically.
Hence this is the component that determines the time of flight.To find the velocity it leaves the ramp (horizontal):GPE lost = KE gainmgh = 1/2mv2gh = 1/2v2v2 = 2ghv = V(2gh)To find the time of flight (vertical):s = ut + 1/2 gt2s = 1/2 gt2t2 = 2s /gt = V(2s / g)Now finally we can use a very basic formula to find the range. We have got the velocity which is determined by the height on the slope where the ball is dropped. And also the time of flight is determined only by the vertical component.
By substituting them together correctly we should be able to get the range.Range = Velocity * Timer = V(2gh) * V(2s/g)r = V2 * Vg * Vh * V2 * Vs / Vgr = 2 * VhsPreliminary TestI did some preliminary trials for selecting the appropriate equipment and setting up the apparatus before planning the actual experiment. This allows me to decide the best equipment, thus increasing the accuracy of my results.The first problem I encountered was bending the rail. It was difficult to make the launch angle flat. In the end I have found another method. Since the table was flat which was measured with a spirit level, I decided to force the flat part of the rail into the table.
This worked extremely well as the entire part was aligned with the table. To double check whether it was perfectly flat, I used the spirit level again and unsurprisingly the bubble of air was right in the middle indicating it’s flat.Once the rail was set and clamped in place, I started my trial run. I have tried using ball of different sizes and found that the smaller one seem to hit something coming down the rail. When I tried the large one, there was no problem at all. I suspect this is because the smaller ball is easily affected by dust particles.
I used a sand tray to measure the landing spot of the ball. At first everything seemed to work out fine. But once I was dropping the ball from greater vertical height, the ball would often roll or even bounce back up. To resolve this, I used a sand tray with more sand making it thicker. As a result the sand would be able to absorb more of the ball’s kinetic energy.Ultimately in my plan I have selected the equipment I felt best for sensitivity and accuracy before I start obtaining my results. Hence the method and apparatus I am using to obtain my results are reasonably accurate and reliable.
ApparatusMethod1. First we have to set-up the apparatus as above. We will need to measure the height of the table from the ground and the height of the slope from the table. By adding the height of the slope from the table to the height of the table from the floor, we could find out the overall drop height mentioned earlier on.
The value I obtained for the height of the table was 0.93m from our preliminary test.2. After getting the apparatus ready, we will start the experiment right away. We will be dropping various heights on the slope, from 5cm to 40cm with 5cm intervals. To do this, we hold the ball according to the height on the rail and release it. It will then travel down and into the sand tray leaving a mark similar to half a sphere.
I will be repeating each drop three times and then taking an average to ensure accuracy.3. We will measure from the centre of the mark to the end of the sand tray closer to the edge of the table. By adding this value we obtain with 15cm, we will acquire the final value of the range.Experimental errors and UncertaintiesThere are superficial errors with the ruler I will be using.
The edge of the ruler I will be using is not 0cm. The 0cm indictor is actually 0.3 cm from the edge. As a result I need to minus 0.3cm from the reading I obtain.
The ruler needs to be placed flat on the surface and touch the edge of the sand tray. Furthermore when I measure the reading, I have to read it off perpendicularly so that it’s more accurate.Since all the rulers we use are to the nearest millimetres, we will have absolute errors occurring. We would have inaccuracies determining the height of slope, the drop height, range and the actual reading.
As a result we would have four areas giving us an error of ?0.05cm. As a result the least and greatest reading of the range we would get is approximately 0.18cm after a simple calculation using the equation we obtained earlier on and some addition. Therefore for every reading we obtain we will have a fixed absolute uncertainty of ?0.18cm. This error is rather small, as the percentage error would not even go beyond 2%.
Moreover this error would further compromise as the range increases.Fair TestIn order for a fair test we must keep all other factors affecting the result constant. The launch angle must be secure and ensured to be flat, as it would affect the time of flight, thus affecting the range. Theoretically it doesn’t matter what size or mass of ball to use, but practically there is resistance, thus we must use the same one every time. As for the rail, the gradient should have no effect either, but different rails might cause different results due to its roughness and other factors and therefore we will use the same one. Finally the air in the room will be very calm as windows will be closed and so there will be little air ventilation.ResultsHeight of slope (m)Actual Range(m)AverageVelocity(m/s)Time of flight(s)PotentialRangePercentageDifference between actual Range and Potential Range(%)Percentageof range achieved(%)0.450.
65,0.640.641.980.480.953367ConclusionStudying graph 1, we can clearly see that as the greater the height dropped on the slope, the greater the range. This is because greater the drop is, the more gravitational potential energy (given by mgh) is converted into kinetic energy (1/2mv2).
As a result, the ball which is treated as a particle, gains more horizontal velocity leaving the curtain rail. For more in depth detail, we can refer back to the plan.It is really common sense that dropping from a greater height on the slope leads to a greater range. However our result differs from the potential range that can be achieved. This is because the way our potential results are calculated uses formula that assumes no energy is wasted.
This includes energy transferred into heat and sound due to friction and air drag. Obviously we know it existed in our experiment, hence we can see that our actual result is lower than the potential range. From the results table we can see that the percentage of the potential range achieved decreases as the dropping height on the slope increases.This situation occurs when the velocity of the particle increases, the counter forces, also termed as retardation force increases which is air drag in this case. It exists because air molecules are colliding into the body causing an exact opposing force. In our situation it is travelling on the rail slope freely due to gravity, thus this means the force in the direction it is travelling which is parallel to the slope is constant (given that gravity is constant).
Since the counter force increases as velocity increases acting in the exact opposite way of the direction of motion, the resultant force of the particle along the plane becomes smaller. As a result, according to Newton’s 2nd law (F = ma), due to smaller resultant force, the acceleration decreases. This has an effect on the range, as the velocity of the ball leaving the slope would be smaller. The calculations we used to find out the potential range omitted these factors.EvaluationIn general the experiment proceeded rather smoothly with superficial problems. The result obtained was very satisfying with the available equipment in hand. However there are several ways we could further improve the accuracy of our experiment.Firstly, our experiment could have been more efficient and accurate.
Very often when the ball lands on the sand tray, it does not leave a clear mark, thus we have to retry it. To resolve this situation, we could have used even thicker layer of sand or simply use the carbon paper. Also from time to time, I have to shake the sand tray to make the sand surface flat. This again wastes time and causes minor errors. Actually, with a proper set up, measuring the range with the carbon paper can be less time consuming and giving us an even more accurate result.
Fortunately this would not have an apparent effect or disagreement on the hypothesis and conclusion we have made.Secondly I have also discovered that the sand tray would move slightly outward when dropping the ball at greater height on the slope. I think this is because at the moment of impact, the ball has a greater horizontal velocity. To resolve this problem, I have to check whether the sand tray is back into its position after impact. As a result, I decide to use carbon paper instead next time when recording the range.Also, when dropping the ball, it will not have been dropped from the exact point every time, as this is virtually impossible.
To overcome this problem, some kind of release mechanism would ensure that the ball is dropped from the same point on the slope every time.All of these have virtually no effect on our result or conclusion at all. Instead there is something we could alter that would have assisted us in concluding the formula. For example, a working condition where air resistance is zero. In other words, working in a vacuum environment.
As a result, the condition we are working in is more similar to the assumptions we have made using the formula. With common sense, we can expect this condition would yield a greater range.With no air resistance, I can say that our actual range would be greater because the velocity leaving the ramp is getting closer to the calculated value using the idea that potential gravitation energy lost is equal to the kinetic energy gain.
Basically, more of the GPE is converted into kinetic energy. As a result when plotting graph2 again with the new conditions, we would expect a steeper gradient or a gradient that is nearly the same as the one we have calculated. Ultimately, it can give us stronger evidence that the formula is spot on, even though the degree accuracy we currently have is sufficient.