Aim: The aim of this experiment is to investigate the relationship of the height of the vertical drop and the horizontal distance traveled by an object when it rolls down a slope and experiences a free fall.Introduction:Ski jump is one of the events in the Winter Olympic.
This sport event involves a steep ramp and a landing zone. The main aim of this sport event is that the skier has to travel as far as possible after leaving the ramp horizontally. It is the motion in the air and the range that the skier travels we are concerned with. This motion is called the projectile motion. The displacement, velocity and acceleration of the projectile are all vectors. The forces of the projectile motion can be treated separately. They can be resolved into horizontal and vertical components. They are independent of each other; that is, neither motion affects the other.
In my experiment, I would create a similar model of the ski jump. I will use a plastic track as the slope and model the skier as a metal ball.Method:Firstly, I will have to set up the ramp in the right position. I am going to bend the ramp into a curved shape.
I will hold up one end of the ramp by the clamp stand and the other end will be placed horizontally on the table. It is very important to place the end of the ramp horizontally because we have to ensure that the initial velocity vector has no vertical component: uy=0. In order to make sure that the end of the ramp is placed horizontally, I will clamp a ruler on the side of the table and the tip of the ruler will prevents the ramp moving forward. I will place a sand tray on the ground for the metal ball to land. I will release the ball in different heights in order to estimate the landing location of the metal ball.
I will measure the distance from the ramp to the mark produced by the metal ball on the sand. We call the range the metal ball travels x. In order to make sure that the only variable in the experiment is the height of the vertical drop, I will release the metal ball at the same point on the curve track. I will also repeat the experiment for 5 times to calculate the average.I would need to decide which point on the curve track I should release the ball.
I will release the ball at the highest point of the track. This is because this can increase the time for the metal ball to accelerate and to leave the track with a higher velocity to produce a more significant mark on the sand tray. I will use a marker pen to mark down the point where I release the metal ball to make it a fair test. This will be the point where I release it everytime. I will change the height of the drop landing area after I repeat each of the level for 5 times. I will change it by putting books and magazine underneath the sand tray. The range of the height should be around 20cm to 80cm. It is usually hard to measure the h2 distance exactly, so it will be more or less ï¿½1cm.
The readings will be insignificant to obtain if the value of h is below 20cm or above 80cm because the time is either too short to measure or the distance is more then from the ground. I will also produce a graph while I am recording my results. A table and graph will also be produced in the end.
Here is a diagram of the apparatus.Another diagram in preventing the end of the ramp moving forwards:Equipment:Equipment:Curved Track (ramp)Metal Ball (Skier)Paper trayBlue TacSand trayMeter rulerMarker penClamp standG-clampsBooks and magazineBoxesPrediction:Since I am using h2 as the variable, I will measure it from the point of the metal ball leaving the ramp. I predict that the bigger the h2 is, the further the ball will travel. Since the motion of the drop of the metal ball should be the same (theoretically, when air resistance is the same), the effect of h2 will have no effect to the horizontal motion.
As h2 gets bigger, the time it takes the metal ball to travel in the air is longer and therefore, a longer distance should be travelled. Since the horizontal and vertical components are independent, we can treat vertical component by different calculations.Therefore I believe increasing the h2 will increase the time the metal ball travels in the air and therefore increase the distance. Figure 1 shows how the projectile motion can be separate into 2 components. The vx and vy indicates the horizontal and vertical velocity. Here are the calculations of the motion in my experiment:Using: s= ut + 1/2at2Substituting values into equation:h2 = 0+ 9.8t2/2Rearranging the formula gives:t = ï¿½[(2h2)/a]The horizontal and vertical components are being treated independently.
As vy is 0, it experiences a free fall. We can prove that h2 has an affect in the horizontal distance by looking at the horizontal motion, and using the equation ?v = ?s/?t. We can substitude the numbers again.Using ?v = ?s/?tux = ?sx/ï¿½[(2h2)/a]uxï¿½[(2h2)/a] = sxIn the above equation, the time it takes is the same as the time in vertical motion and since the velocity of the metal ball leaving the ramp will always be the same, it is the time that varies.
Since changing the h2 will have an affect in time, I believe it will affects the whole vertical and horizontal distance travelled.So far what I have predicted is based on theory and calculations. In the real world, there are air resistances that oppose the action of the metal ball. It will act on both vertical and horizontal motion and will reduce the distance it travelled. The distance will varies and this is the reason for me to take the average.Adjustment:A pre-test is done before the actual experiment and I would like to make some adjustment of my apparatus. Due to the different sensitivity of the equipment, I would not use the sand tray. I would change to carbon paper.
By using carbon paper, I will put a cardboard under the A4 paper and place the carbon paper on top. This helps to make a more significant mark on the paper. Using the sand tray is not a good idea because the metal ball rolls on the tray after it hits the sand and created more then one mark for me to measure and it is very inaccurate. I will use a big size metal ball which is heavy enough to make an obvious mark on the paper. I am going to use regular sizes boxes to increase the h2.
Because of the nature of the soft surface on the boxes, the metal ball is not able to make an obvious mark on the A4 paper. This is the reason for me to place a cardboard under the paper to make it land on a hard surface. For the plastic ramp, it bends slightly after it is being clamped and before each recording, I would stick blue tac under the ramp and make sure that the end is horizontal and the horizontal distance is the same for every result. This distance will have an affect on the launching velocity. The launching height will be 20cm to 80cm, from the pre-test; I believe it gives a reasonable range to measure the distance.Safety Precautions:The metal ball I am using weighs around 45 grams and it can injure people walking pass by if being hit.
I need to make sure the landing zone is clear of obstacles. I will make sure there are no breakable objects around because the metal ball will bounce for a distance before it stops.Tables of result:Vertical distance from the horizontal launching level (h2) ï¿½1.
0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.0 (uncertainty) x 10020.0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result20.067.85%1.
46%Average:68.085%1.46%Vertical distance from the horizontal launching level (h2) ï¿½1.0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.
0 (uncertainty) x 10030.0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result30.071.
37%Vertical distance from the horizontal launching level (h2) ï¿½1.0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.
0 (uncertainty) x 10040.0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result40.075.
29%Vertical distance from the horizontal launching level (h2) ï¿½1.0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.0 (uncertainty) x 10050.
0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result50.081.42%1.22%50.
22%Average distance:81.642%1.22%Vertical distance from the horizontal launching level (h2) ï¿½1.
0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.0 (uncertainty) x 10060.0Percentage Error (%) for range1.
0 (uncertainty) x 100Actual Result60.085.71.67%1.16%60.085.61.67%1.
16%Average distance:85.641.67%1.16%Vertical distance from the horizontal launching level (h2) ï¿½1.
0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.0 (uncertainty) x 10070.0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result70.089.91.43%1.
106%Vertical distance from the horizontal launching level (h2) ï¿½1.0cmTotal horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for (h2)1.
0 (uncertainty) x 10080.0Percentage Error (%) for range1.0 (uncertainty) x 100Actual Result80.
07%Average distance:93.121.25%1.074%Taking out all the average distance from the tables and here is a final table of what I have got:Vertical distance from the horizontal launching level (h2) ï¿½1.0cmAverage horizontal distance travelled (range) ï¿½1.0cmPercentage Error (%) for range1.
0 (uncertainty) x 100Actual ResultTotal Error %(Sum of error in h2 and error in range)20.068.081.46%6.46%30.072.681.
074%2.324%Graph:I have produced 2 of the same graphs. One is computer generated and one is hand-drawn. The hand-drawn shows different interval of the horizontal distance traveled. The computer-generated graph shows a more accurate and precise result from the table. The results in both graphs are the average horizontal distance traveled by the metal ball:Analysis:As you can see from the graph, as h2 is increasing, the horizontal distance also increases.
The graph is more or less a straight line because the horizontal distance travelled by the metal ball in each interval should more or less around the same. However it didn’t show a perfect straight line. This is because of the inaccuracy of the equipment, measurement and limitations (air resistance).
The meter ruler and the boxes have its thickness and this create another problem in measuring the height of landing. Therefore the uncertainties would be the thickness of the boxes. It will be more or less ï¿½1cm.The position for releasing the metal ball is another issue. If the ball is being released from a higher or lower position, it will have an affect in the initial velocity leaving the ramp. Higher velocity will result in a bigger horizontal component and therefore uncertainty would be more or less ï¿½1cm. The total uncertainty would be ï¿½2cm.
However, the error bars are so small that it didn’t show up very obviously in the computer-generated graph.Another important factor is the piece of A4 and carbon paper I used to measure the horizontal distance. It is not stable and will relocate after the metal ball hits and bounces off.
The percentage error of h2 and the range is calculated and they are shown on the table. The error is not very big, however it can still be eliminated. This will be discussed in more detail in evaluation.The plastic ramp also creates a problem. The end of the ramp is difficult to maintain horizontal because of the stiffness of the ramp and it is slightly bend. This is an important factor because it has an effect on the initial velocity and therefore will change the results. It will create a vertical acceleration if it is bend. (vy >0)In theory, we have assumed that the air through which the projectile moves has no effect on its motion, a reasonable assumption at low speeds.
However, for a greater speed, the disagreement between calculations and the actual motion of the projectile can be large because the air opposes the motion. Therefore, for bigger h2 being in the air longer (time is longer from calculations in prediction), the air resistance will oppose the action longer and therefore will reduce horizontal distance travelled. I believe this is the reason that the line on the graph bends slightly. By calculating the differences, we can see that the differences get smaller because air resistance acts on the metal ball longer, as a result reduced the total horizontal distance travelled. Therefore in the absence of air resistance, I believe that the graph produced would be a straight line.
Figure 2 shows the projectile path of the metal ball in an ideal and real world: and this proves air resistance is one of the biggest limitations in this experiment.I have calculated the differences in horizontal distance for each interval from my results and the theoretical interval. I am assuming the theoretical horizontal distance travelled is the same and take the first result as the distance. I also assume that air resistance is neglected, so there are no forces opposing the action of the projectile and therefore leads to same distance in each interval.h2 value (cm)Actual horizontal distance travelled (cm)Theoretical horizontal distance travelled (cm)20.04.64.
964.6However, the actual horizontal distance travelled from my results shows that; the distance is gradually decreasing. It is the air resistance that oppose on the action of the projectile. The metal ball stayed in the air longer, so resistance force is bigger. However, for results taken at around 20-30 cm, the time it takes the metal ball to land is almost the same, therefore air resistance only have a small effect.Surprisingly I found that the difference between 40 to 60cm has a bigger gap between them and I believe it is an anomalous result because it decreases down a big gap. It decreases by 0.
44cm and then by 1 cm. I think this is because of both the sensitivity of the equipment and air resistance. (Highlighted in red)Looking at the percentage error of my results, they are not a big value there won’t be much effect on the whole curve in the graph. This is the reason why the computer-generated graph cannot shows the error bars clearly.Another point is the rolling effect of the metal ball.
I believe this has an effect on the velocity and the range travelled. The differences between rolling and sliding effect will change the projectile motion in terms of friction. Since rolling down will constantly change the contact area with the ramp, this will also change the friction and therefore result in higher initial velocity. However, if the object is sliding down, then the area contacting the ramp will be constant and therefore friction will be the same. Assuming air resistance is ignored, I believe the only difference between rolling and sliding is the initial velocity produced. This will result in bigger and smaller projectiles. The following diagram shows the force acting on the metal ball.
Evaluation:There are different things that I could change to improve the accuracy of the experiment. I can put a gate as a releasing barrier at the point of dropping the metal ball to ensure that it is being released at the same point everytime. This can also help to keep the initial velocity leaving the ramp the same. Strong tapes and sticky blue tac can be used to keep the end of the ramp in position and to maintain at horizontal level. This helps to provide uniform acceleration for the vertical component.If possible, I can also use portable stands where it can moves up or down at certain level so that I can eliminate the uncertainty produced by the boxes. It can change h2 effectively and reduce the percentage error to the very limit. Tapes can be used to stick the carbon paper in place as well.
There is a way for me to measure the initial velocity of the metal ball leaving the ramp. I can place a light gate at the end of the ramp to measure the time for the ball takes to travel. Then measuring the diameter of the ball gives me the distance and I can use the equation ?v = ?s/?t to calculate the initial velocity. After that, substitute the initial velocity into the equation s= ut + 1/2at2 (As I have explained in my prediction) can gives me the initial velocity leaving the ramp (ux).
Using ?v = ?s/?tux = ?sx/ï¿½[(2h )/a]uxï¿½[(2h )/a] = sxTherefore I can use it to calculate a more precise value of the theoretical distance travelled by the metal ball. It can provide more evidence for me to investigate the actual horizontal distance traveled.In order to make the results as accurate as possible, we can try to eliminate all the uncertainties. However, some of the limitations cannot be eliminate. Air resistance is always present and is a force that always opposes the action of the metal ball. However, I can change the acting of the metal ball.
I can use object that slides or rolls down. However, it is hard to make the same object slide and roll down at the same time. The metal ball is always going to roll down and produce a spinning effect.
If I use another object to slide down, the air resistance would be different and therefore it will only result in increasing the uncertainties. But I can still carry further investigation in these criteria. Another interesting point would be changing the vertical unitform acceleration of the metal ball. However, I believe it is impossible for me since we will have to change the gravity of that area for the metal ball to accelerate faster then 9.
8ms-2.In this investigation, I have taken account the h2 as my variable and all other factors are being fixed. I can also investigate h1 if I will do this experiment again. It is different from h2 because the difference in launching height gives different initial velocity and therefore gives different horizontal distance travelled.Conclusion:The conclusion to this experiment is that the vertical projecting level increases with the horizontal distance traveled and the horizontal component should always be the same in theory. However, it is the air resistance that limits the projectile motion.
From my results, it shows that air resistance only has a little effect on the metal ball if the time is less. However, if the metal ball travels in the air longer, the air resistance opposes the action longer and affects the horizontal distance traveled. It also proves that air resistance is a big limiting factor. This is the reason that makes the line on the graph bend slightly.
It starts to bend at the point of 50-60cm. This turning point shows the effect of air resistance and the horizontal distance also starts to decrease at a bigger interval at that point.