Crows dropping nuts

Task:

- 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.

- Data table given:

- 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. However, the data presented here is an average over a large number of trials, and because of this n can be any positive number, either an integer or fractional. Now another point to note about the value of n is that it represents the ‘average’ number of drops over a number of trials for different size nuts. This is assumed to imply that n is the mean of the data set for each value of h. However if the data set was not large enough the mean may not be a precise measure, and error bars should be associated with each value of n because of this the functional fit, which may represent the actual value of n, f(h) may not lie on the data points given, and this may go to partly explain any variations between the fitted curve and the data points, even if it is only a small effect.

The model given for both h and n will be the fit for discrete data, whereby there is a variation in the variables. It is noted that the table record containing the average number of drops is only a measure of the height and number of drops required to crack nuts, whereby a number of variable such as size and mass of the nut are not presented. Therefore it is expected these differences from trial to trial would account for variation in any model attempting to fit the data, as these would have significant contributions towards the nut cracking from different heights. As well as it was also not stated if the craw was stationary when dropping the nut, therefore in the investigation it was assumed that the craw dropped the nuts while stationary.

This can also be explained by standard deviation, which can be used to calculate the average or most likely outcome. However, there is always a range of values that can reasonably attribute to the results. Whereby low data values indicate that the points tend to be very close to the same value and the high data points are ‘spread out’ over a large range of values, this known as standard deviation as shown below. (Citation)