The Physics behind Ski Jumping

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					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


       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

                                                               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

                                         Agrav  9.8      .

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

                                                y y  y0 y  a y t 2
                                       v0 y                2        .                Equation 2

        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.

                      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


                          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

                          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:

                                     x(m) v 0 x ( )t ( s)  x0 x                    Equation 3

                                     x  4.71     x  10.5m                       Function 1.3

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


                                              x  5.01     x  11.4m              Function 2.3

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

                                  x  4.71( ) x  10.5m                         Function 3.3

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.


       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

gravity, 9.8      , were visibly smaller then expected. The only way that we would be able

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


VideoPoint human inertial mass
      Priscilla Laws and Mark Luetzelschwab oct. 1995
      Obtained by Zatsiorsky et al

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