Document Sample

The Physics behind Ski Jumping Written by: Matt Polichette Jenn Demers Angela Marler-Boyd Mitch Vernon November 21, 2006 Physics 211 Section 1 Fall Semester Project The Physics behind Ski Jumping 2 Abstract Our project was designed to calculate how the initial velocity off a ski jump affects the parabolic arc, which the ski jumper follows through the air. We also want to show how projectile motion applies to ski jumping. We were able to use VideoPoint to obtain values for velocity and acceleration and using the kinematic equations we could determine the velocity required to complete their specific jump. We’ve also shown how the horizontal velocity of the skier remains constant after he leaves the jump while in the vertical direction, and acceleration stays constant. The Approach In our project we took up the analysis of freestyle ski jumping. Freestylists perform aerial acrobatics that cause them to rotate and flip multiple times, while traveling in a parabolic curve. All jumpers use the same ramp in order to obtain an optimal velocity before leaving the jump. This velocity allows a suitable height for aerial tricks. If the velocity coming off the ramp was too great the ski jumper could easily be injured by the impact, due to gravitational acceleration. In the opposite case, if the velocity off the ramp was too small the skier would not have enough time to complete the aerial trick and would not be able to land flat on their skis. Our attempt is to apply projectile motion physics to the movement of the aerialists. The jumpers use their airtime to execute their acrobatics, we analyzed how the jumper’s initial velocity at the base of the jump affected the height of their trajectory and The Physics behind Ski Jumping 3 thus the amount of time they spent in the air. At first glance their motion may appear skewed because of the rotations and flips but using the concept of center of mass, (CoM), we will show how the laws of physics continue to hold strong. Ski Jumping 101 Freestyle skiers spend a large amount of time perfecting their form and jumps. They build speed going down a hill or, during the summer, a water ramp which allows them to launch off the jump and perform aerial acrobatics. Their sport requires precision in timing, and if they don’t achieve just the right velocity when leaving the jump, they’re in for a world of hurt. If they go too fast they will overshoot the landing and land flat. If they undershoot the landing, by going too slow, they may not have time to complete their flip or rotation, possibly causing them to land upside-down. Like any object experiencing free fall with an initial horizontal velocity, Freestylists can be analyzed using projectile motion physics. By calculating changes in skier’s position over time we can find velocities and accelerations, and knowing angles of jumps we can tell just how far and how high the skier will travel. These are important because the further a skier may fall the more acceleration he experiences, and the construction of his landing will need to enable him to complete the landing without causing injury. The Equipment and Methods The Physics behind Ski Jumping 4 We used video’s of ski jumpers at Park City Olympic Park and VideoPoint to calculate the jumpers position, velocity, and acceleration in both the x and y directions. We decided to place our origin at the point where the skier lands in the water so the skier is always above the x axis. In order to scale our movie, we found a portion of the ramp that was equal to 1 meter and that was visible in all of the movie clips. By scaling we will be able to use our points in meters as opposed to pixels. In our video each part of the body moves at a different rate, which makes calculating the entire body’s vertical acceleration difficult. Though, using the distribution of human mass provided by VideoPoint we were able to choose 8 significant points on the skiers body and assign a percentage of mass to them: Location on Body Weight percent Trunk 42 Head 6 Left Arm 5 Right Arm 5 Left Leg 18 Right Leg 18 Left Ski 3 Right Ski 3 CoM 100 Table 1 Collecting the location for each of the 8 points in each frame, we were able to calculate the center of mass of the skier throughout the jump. Scaling the movie produces the position coordinates of the skiers CoM in each frame, and by applying kinematic equations we can determine velocity and acceleration throughout the jump. VideoPoint also allows us to create graphs that represent this information and analyze them simultaneously. The Physics behind Ski Jumping 5 Because we’re dealing with projectile motion and constant gravitational acceleration we can use the kinematic equations to model the path the object follows and calculate values for velocity (v), and acceleration (a), using position (x) and time (t). The third kinematic equation was most useful for our situation: m 1 m y (m) y 0 (m) v0 y ( )t ( s) a0 y ( 2 )t ( s) 2 Equation 1 s 2 s We used this equation to find the resultant velocities and acceleration for each jump with the data collected through VideoPoint. Once the skier leaves the ramp they begin to experience the negative acceleration of gravity, which is a constant (close to the earth) of m Agrav 9.8 . s2 Because our skier starts above the origin, we have to find the initial height above the water level. We find that value by placing a point at the skier’s center of mass at the tip of the jump and measuring the distance to the x axis. We know that at the maximum height the skier’s velocity will be zero. Using VideoPoint data collected from each of the frames, we can find the time when the CoM is at that point. After rearranging equation 1 we can easily solve for the initial velocity; The Physics behind Ski Jumping 6 1 y y y0 y a y t 2 v0 y 2 . Equation 2 t Putting the values for max height ( y ), initial height ( y o ), acceleration ( a ), and time ( t ) into equation 2, we were able to calculate the initial velocity of the skier leaving the ramp in the y direction. Applying trigonometric equations to the y o we can determine the values for the initial velocity relative to the angle and x velocity off the jump. VideoPoint allows us to graph the x positions against time in order to check our calculated results against experimental points. In order to calculate the amount of time each skier spent in the air, we used the tables we created in VideoPoint, taking the first point as the skier was leaves the ramp, and the last point taken as the skier hit the water, our origin. The difference between the time values for each of these points is equal to the amount of time in the air. We can use equation 3 to find a calculated velocity ( v y ) as the skier hits the water by entering initial velocity ( v 0 y ), acceleration ( a grav ), and the total amount of airtime ( t ). v y v0 y a grav t Equation 3 Sticking the Landing For our first jump we analyzed a skier performing a corked 720 and started taking the data point’s right as they left the jump. First we calculated a distance of 2.9 meters, between the x-axis and the skier’s CoM at the tip of the jump. Then using the procedure outlined above, we created graph 1 for the y position of the skiers CoM: The Physics behind Ski Jumping 7 Graph for y Jump 1 Using the line fit function we created an equation that fits our data and represents it in the form of the third kinematic equation. 1 x(m) x0 y (m) v0 y (m )t ( s) a0 y (m 2 )t ( s)2 s 2 s Kinematic eqn. y .99 (m) 10 .7(m )t 4.43(m 2 )t 2 Function 1.1 s s Our data fit fctn. We find that our line fit equation gives us a gravitation acceleration of –8.86 m/s2 instead of 9.8 m/s2. We decided to use this measured acceleration to find the initial velocity off the jump. By calculating a distance of 5.52 meters at the peak height of the CoM at 1.23 seconds and entering our data into equation 2 we get an initial velocity in the y direction of: m 5.52m 2.9m (4.43 )(1.23s) 2 s 2 m v0 y 7.59 Function 1.2 1.23s s Trigonometric functions of the cos and tangent allow us to find our other velocties: The Physics behind Ski Jumping 8 v (m ) vx ( m ) 0y s s tan(305) *305 is the angle from the top of the jump to the vertical component Graph 2 below is of the second jump the graph consists of data taken as the skier exits the ramp. In this jump the skier does a straight back flip. Graph for y Jump 2 This is the kinematic equation fit line for graph 2. y .193 (m) 10 .8(m )t 4.67 (m 2 )t 2 Function 2.1 s s m 3.55m 5.46m (4.43 )(1.13s) 2 s 2 m v0 y 7.49 Function 2.2 1.13s s Graph 3 below is an analysis of a skier doing a corked 720, like graph 1. The Physics behind Ski Jumping 9 Graph for y Jump 3 y .076 (m) 9.83(m )t 4.41(m 2 )t 2 Function 3.1 s s 2.75m 5.46m (4.41m 2 )(1.13s) 2 v0 y s 7.38 m Function 3.2 1.13s s Also graphing the velocity in the x direction we were able to see how in conforming to projectile motion, the velocity remains constant and there is no acceleration through the entire jump. This is true because there are no large forces acting in the x direction after the skier leaves the jump. Graph 4 below shows the constant velocity by the slope of the position vs. time for Jump 1. The Physics behind Ski Jumping 10 Graph for x Jump 1 As one can easily see, the correlation of the points follows a linear approximation. The form that represents this line is: m x(m) v 0 x ( )t ( s) x0 x Equation 3 s m x 4.71 x 10.5m Function 1.3 s Looking at this equation it is obvious that acceleration does not play a part in the movement in the x direction. Knowing this we can see that velocity is a constant value throughout the jump in the x direction and that no forces are acting on the skier in the x direction, therefore nothing is causing acceleration. Graph 5 below is the position vs. time graph in the x direction for the second jump. m x 5.01 x 11.4m Function 2.3 s This graph shows a velocity that is larger then the value calculated for jump 1. Again the data fits a linear equation with no acceleration over the jump. Graph 6 below is the x position graph for the third jump. The Physics behind Ski Jumping 11 m x 4.71( ) x 10.5m Function 3.3 s The line that best fits this data is the same line as the equation present to represent the first jump, which was of a similar style of jump, just a different skier. Judging the Jump Using our graphs and our line of best-fit equations we were able to find values for acceleration, velocity and position, over a specific amount of time. Equations were created in both the x- and y-axis. The r2 values for the line of best-fit equations represents how accurate the model of the data points are. All of our graphs, besides one graph of the x velocity, have an r2 value equal to 1 which means all of our data points fall exactly onto the line of best-fit equation provided by VideoPoint. The Graph of X jump 2 with the r2 value less then one has a value of .999, which is very close full accuracy. When collecting data points we realized that the rate at which the skiers were falling during the freefall were not exactly equal to the rate of gravitational acceleration The Physics behind Ski Jumping 12 or 9.8 m/s2. This decrease in the acceleration may be caused by the air resistance against the skis and the skier as well as other forces acting in opposition to the force of gravity. And the crowd goes wild! Conclusion here………………………> and here (-)^ Jump 3 had the smallest value of initial velocity but had a maximum height of CoM very comparable to Jump 1 which had completed a similar trick. All of the ski jumpers hit a maximum height and started free fall at different times in their jump. Jump 3 had the shortest time by .15 seconds. By comparing the 3 skier’s maximum height obtained in the y direction with the total displacement in the x direction, we can see if a correlation exists. We can also see if The Physics behind Ski Jumping 13 there is a correlation between their initial velocities in the x direction before they leave the ramp and their maximum height in the y direction. Technicalities There were several points of uncertainty in our calculations, one major discrepancy can be found in the distribution of weight, while our calculations are generally accepted they cannot be assumed to be exact for all individuals. The skis of the aerialists also affect their center of mass by adding more weight to the jumper on their bottom half, without accurate measurements of the mass of the skis; comparable to the mass of the jumper we could easily be measuring and collecting data on the wrong center of mass. When scaling the movie in video point it is quite possible that our scale varied by a few pixels, which could lead to varying data throughout the various movies. The scale can also be affected by the fact that the movies were taken from different distances and heights from the ramps and jumpers. Pixels do not translate to different depths, which would be necessary for calculating, the clips we are analyzing have been moved into a one-dimensional plane. The Physics behind Ski Jumping 14 Our graphs and equations were created from data that may or may not be accurate. By using the center of mass, we were able to calculate the equations to that point, however center of mass on a person shifts, especially when said person is flipping through the air, and changing their center of mass instantaneously. To fix these uncertainties in the future we can use a tripod for the camera and film from a set distance, parallel to the ramp for every jump, also by taking a scale that is at the same depth as the jump. If we were to calculate the center of mass of the individual jumpers with their skis attached before the jump, we could mark their individual center of mass, and be able to follow that through the parabolic shape. Our y accelerations, which should have been equal to the accepted value of m gravity, 9.8 , were visibly smaller then expected. The only way that we would be able s2 to achieve a y acceleration that is equal to gravity is to place the skier into a vacuum, which is not possible in this situation, therefore, we must account for all forces. The Physics behind Ski Jumping 15 Calculating the initial velocity on the ramp gives us a value that we can compare to the height and observe if there may be a correlation. Bibliography VideoPoint human inertial mass Priscilla Laws and Mark Luetzelschwab oct. 1995 Obtained by Zatsiorsky et al

DOCUMENT INFO

Shared By:

Categories:

Tags:
Ski Jumping, surface area, Ski Jumps, ski jump, air resistance, ski jumper, diffuse reflection, first place, Ski team, ski wax

Stats:

views: | 782 |

posted: | 2/19/2010 |

language: | English |

pages: | 15 |

OTHER DOCS BY tyndale

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.