• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15
  16. 16
  17. 17
  18. 18
  19. 19
  20. 20
  21. 21
  22. 22

Crows dropping nuts

Extracts from this document...


Crows dropping nuts


  1. Introduction:

Crows love dropping nuts but their beaks are not strong enough to break the nuts open. To crack open the shell they will repeatedly drop the nut on a hard surface until it opens. In this investigation we will determine a possible function that would model the behaviour of the data given below.

  1. Data table given:

Height of drops (m)











Number of drops











  1. Definition of variables as well as constraints:
  • Variables: Representing the Number of Drops and Height of Drops,

Let ‘h’ be the height above ground (in metres) and ‘n’ be the number of drops favourable outcome (nut crack). Whereby ‘h’ is the independent variable and ‘n’ is the dependent variable. Therefore all graphs will be plotted as ‘h’ (height of drops) and ‘n’ (number of drops).

Another variable that should also be accounted for is the size of the nut ‘Sn’. It was stated that the average number of drops will also be investigated using medium and small nuts. Therefore ‘Sn’ will be used to illustrate the size of the nuts.

  • Constraints: Representing the boundary values and types of numbers for h and n,

h is a positive integer such that: 0 < h. Height is a displacement measure, it tells you the vertical displacement of an object from a ‘ground’ position. For this data it is assumed that h = 0 is the ‘hard surface’ whereby the nut impacts.

 n ≥ 1 because it must take at least one drop for a nut to successfully crack from a very high height. For one particular trial the number of drops it takes for a nut to break must take an integer value. For example, the nut cannot break on 1.4 drops; it would either break on the first or second attempt.

...read more.



Thus after having calculated the motion of the nut under constant acceleration, therefore it was also important to calculate the momentum of the nut, as this is the quantity that is conserved under an inelastic collision.  (Citation http://en.wikipedia.org/wiki/Momentum)


p = momentum      

m = mass

v = velocity

p = (kgms-1)

m = (kg)

v = (ms-1)

Hence knowing the velocity of the nut from the previous equation, the following was deduced;


Thus it was assumed that the impact of the nut will be proportional to the momentum,


However as the nut would bounce off the surface not all the force of the impact with the ground go into deforming the nut (breaking) (Citation http://en.wikipedia.org/wiki/Deformation).

Therefore another force is introduced,

image12.pngimage12.png  where image13.pngimage13.png represents the percentage of force that is absorbed by the nut on impact, and is why the notation image13.pngimage13.png is chosen as it is the constant of deformation in this model. Hence the range is:image14.pngimage14.png.

The following is a representation of the formula used to calculate the impact of the nut, whereby, image15.pngimage15.png reflects the force of impact. Therefore by substituting the previous expression for momentum ( image17.pngimage17.png ) into the formula for ‘deformation of the nut’. The next step was to calculate the required number of drops for the nut to crack.

image18.pngimage18.pngBut, image19.pngimage19.png

Therefore the image20.pngimage20.png = image21.pngimage21.png will be substituted for image23.pngimage23.pngin the equation below,


It is evident that a nut will break when it is exposed to enough impact force. There will be a threshold were after sufficient impact force the shell cannot contain the energy caused by impact and will fracture to conserve energy.

image25.pngimage25.png where by image26.pngimage26.png

For each drop the nut will get weaker (i.e. more deformation to its atomic/molecular lattice). Hence the following equation is a representation of the number of impacts,


Therefore when,  image28.pngimage28.png

...read more.


It can be assumed through the results above that so long as the data being collected maintains an accurate curve with minimal variation; the reciprocal power function will fit the data quiet accurately. However if there was variation within the data as found in the small nut, hence the equation will not correctly fit the data.  However in the first equation (reciprocal function) it was proven that the model was not an accurate fit to the data. Whereby the curve of the data resulted in having a negative value. Hence the equation needed to be modified for a better fitEven though after modifying the data and forcing a horizontal asymptote to better fit the data points it appeared that the model still did not fit the data accurately, and hence the equation needed to be changed.However if all the parameters were modified as shown in the graphs above it appeared that the equation fits the data quiet well. However this equation has many limitations. If the curve of a particular data (other than the ones investigated) had a steeper slope thus the equation would not obtain an accurate fit. Moreso if there was more data calculated, thus it would be safe to assume that the equation after a certain amount of time would completely miss the data, because it apparent that the curve is constantly descending, thus after a specific period of time the curve will cross into the negative value. Whereby the equation has reached its utmost limit.  
  • However as for this investigation it can be concluded that a reciprocal function is not as accurate as a reciprocal power function.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Maths IA Type 2 Modelling a Functional Building. The independent variable in ...

    40 100 50 72 2.5 9000 3600 41 102.5 44.721 72 2.5 8049.845 3219.938 42 105 38.730 72 2.5 6971.370 2788.548 43 107.5 31.623 72 2.5 5692.100 2276.840 44 110 22.361 72 2.5 4024.922 1609.969 45 112.5 0 72 2.5 0 0 Total: 795688.274 318275.310 Now, after determining the total

  2. A logistic model

    35000 30000 25000 20000 15000 10000 5000 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Year Figure 4.3. Growth factor r=2.5. Graphical plot of the fish population of the hydroelectric project on an interval of 20 years using logistic function model {9}.

  1. Maths portfolio Crows dropping nuts

    therefore showing a clear limitation to the graph. Whilst the shows that it has two asymptotes just as the original graph does, therefore portrays it more accurately. Another difference between the graphs is their steepness, where the inverse exponential is less steep then the therefore it does not cover the top half of points on the original graph.

  2. Modelling the amount of a drug in the bloodstre

    It shows a regular steady increase but in fact realistically this assumption is incorrect because after 6 hours the amount of drug remaining from the previous drug intake is added to following drug intake. So hence this means that the amount after 6 hours would include residual amount of the previous doses.

  1. Investigation of the Effect of Different types of Background Music on

    can cause major distractions and noise can have a negative effect on students' concentration. The Encoding Specificity Principle of Memory (Thompson & Tulving, 1970), states that the most effective way to recall information is to study the material under the same conditions as when it is to be recalled.

  2. Body Mass Index

    The graph modeled from the values I worked out and the graph modeled from original data 4. Use technology to find another function that models the data. On a new set of axes, draw your model functions and the function you found using technology.

  1. This essay will examine theoretical and experimental probability in relation to the Korean card ...

    is split into eight different possibilities Graph 7. Tree Diagram of Ddang-4 = 4.9545604 x 10-3 Probability of winning with March (Ddaeng-3) is split into nine different possibilities. Graph 8. Tree Diagram of Ddang-3 For Ddaeng-3, it is slightly different than other hands.

  2. Population trends. The aim of this investigation is to find out more about different ...

    The actual model isn't very similar to the data and it can't be changed by a lot in terms of the shape that the curve has. This model of equation is very similar to the data from the year 1960 to the year 1990, the other years are very far

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work