Siddharth RajendranIB Physics HL II28th February 2013Physics Internal AssessmentResearch Question: - What is the effect on the diameter of the crater formed by dropping a metal ball from different heights?Background Information:- It has been observed that whenever a solid spherical object is dropped into a sand surface , a circular crater is formed due to the impact. The diameter of the crater formed is related to the kinetic energy contained within the ball. In this design lab, a metallic ball with a constant mass m will be dropped from different heights and the relation between the heights from which the ball is dropped h and the diameter of the crater formed D will be investigated.

In order to establish a relation between the height and the diameter of the crater, it is necessary to have some pre requisite knowledge about the relationship between the K.E of an object on the diameter of the crater.

The effect of the diameter of the crater due to the K.E of a meteor is given by

Eq. 1

Where: D = Crater Diameter Cf = Collapse factor of the Crater ge = The acceleration due to gravity at Earths surface g = Acceleration at the surface of the body on which the body is formed. E = The kinetic energy of the mass ( i.e the meteor ) a = Density of the impacting body. t = Density of the target rock. Since this experiment has been conducted on earth, ( ge = g ). Also Cf, a , t are all constants and are unknown. Therefore, by taking all these factors into account, the equation can be simplified to Eq. 2 Where k is a constant and may be calculated.

As the metal ball falls from a certain height, its gravitational potential energy will be converted into kinetic energy due to the law of conservation of energy which states that energy can be nor created or destroyed but will only change its form. Thus, K.Emax = G.P.Emax (ignoring negligible energy lost due to air resistance). The kinetic energy of the ball can be calculated by using the formula ( K.Emax = mgh ), where m is the mass of the metal ball, g is the acceleration due to gravity and h is the height above the sand from which the ball is dropped. By simplifying the equation: Eq. 3

According to equation 3, the diameter of the crater is proportional to the height raised to the power of. It is not sure whether the exponent will be applicable to this experiment as the 1st equation applies to meteor collisions on an astronomical level. However the relationship between the crater diameter and the kinetic energy should be a power relationship, in which case, the constant k1 and the exponent of the power relationship denoted as n hereinafter will be calculated.Hypothesis: -An increase in height from which the metal ball is dropped from will result in an increase in crater diameter proportional to the height raised to the nth power where n is an unknown constant, with the form D = k1hn. This is because the previous equation, which has been derived from equation 1, predicts that there will be a exponential relationship between the height and the diameter of the crater.Variables: - 1. Independent Variable The height from which the metal ball is dropped (meters) as measured from the surface of the sand bed.2. Dependent Variable The diameter of the crater (centimeters) formed due to the impact of the metal ball on the sand surface.3. Controlled Variables Mass of the metal ball (225.373 g 0.001 g). Diameter of the ball ( 3.8 cm 0.2 cm )Note: - The following are confounding/extraneous variables:- Air friction ( This is because increased height increases the air resistance ) Initial spin of the ball (The spin of the ball can change the shape of the crater, resulting in a non-cyclic crater ) Average size of the sand grain ( It is unlikely that the container filled with sand particles is of similar size and shape )

Apparatus: - Metal Ball ( 225.373g 0.001 g ) Sifter Circular container Electronic balance ( 0.001 g ) Clamp Meter Stick ( 0.001 m ) Two plastic Rulers ( 0.1 cm ) Magnifying Glass Logger Pro Vernier CalliperProcedure: -(Measuring the diameter of the crater created in sand by ball dropped from different heights)1. The container was filled with approximately 5 centimeters of dry sand.2. The surface of the sand was levelled by gently shaking it. Care was taken not to compress the sand.3. The meter stick was attached to a burette clamp on a retort stand such that the end point of the meter stick was in level with the surface of the sand. The meter stick was used to measure the height of the ball.4. The mass of the ball was recorded on an electronic balance.5. The diameter of the metal ball was measured using a Vernier Calliper.6. The metal ball was dropped at an arbitrary height between 10 cm and 100 cm above the surface of the sand.7. The metal ball was removed carefully after the impact on the sand had been made using tongs so that the borders of the crater formed were not disturbed.8. Using a plastic ruler, the diameter of the crater formed was measured.

9. Using the other plastic ruler, the surface of the sand was leveled out.10. Steps 6-9 were repeated for 7 different heights.

Raw Data: - Crater Diameter (cm0.2cm)

Height(m0.005cm) Mass of ball (g0.001g)Trial 1 Trial 2 Trial 3 Trial 4 Trial 5

1.000225.37312.511.711.711.512.0

0.900225.37311.211.511.711.511.6

0.800225.37310.810.210.310.410.9

0.700225.37310.210.610.110.19.6

0.600225.3739.39.29.79.49.5

0.500225.3738.18.48.18.58.7

0.400225.3738.08.78.68.28.5

0.300225.3737.78.08.37.48.2

Sample Calculations: - Average Crater Diameter

Error Propagations 1. Relative uncertainty for height: - This was calculated by dividing the absolute uncertainty of 0.005 m by the height, then the result was multiplied by 100 %.

2. Absolute uncertainty for average diameter: - This was done by dividing the range of the diameter of 5 trials by 2.

3. Relative uncertainty of the avg. diameter: - The relative uncertainty is calculated by dividing the absolute uncertainty of the average diameter by the average diameter, then multiplying the result by 100%.

Processed Data Table: -

Graphs: - The graphing utility used for this experiment is Logger Pro. On the X axis, the height is plotted (independent variable). On the Y axis, the values of the crater diameter are plotted (dependent variable). In order to plot error bars for the graph, uncertainty bars are plotted for the crater diameter because the value of the largest relative uncertainty of the diameter is greater than that of the height (5.68 % vs 1.67 %).

The following graph shows the relationship between the height and the diameter of the crater.

For the data, power fit was applied. The line of best fit passed through all the error bars, suggesting high reliability of the collected data. In the form of D = k1hn, k1 was calculated to be 11.66 0.1820, and n was calculated to be 0.3641 0.0292. Thus the line of best fit suggests that: - D = 11.66h0.3641

The root mean square value (RMSE) value for this graph is 0.2913, which suggests that the correlation is relatively strong. However, it seems that the data followed a more linear relationship than a power relationship. This can be shown on the following graph, in which linear fit was applied.

As it can be observed, the RMSE value for the linear fit is 0.1536, compared with that of 0.2913 which was a power fit. This indicates that the data followed a more linear relationship than a power relationship. In order to confirm the hypothesis, I collected two more data points with heights 0.1m and 0.2m. The extra data points collected are as follows:- These extra data points were also added to the original graph. The following graph incorporates both the old data points and the new data collected.

Relationship between height and crater diameter

With the additional 2 data points plotted, the hypothesis of linear relationship has been confirmed. The hypothesis is further supported by the RMSE value for the linear fit which 0.1681. Thus it can be concluded that the relationship between the heights from which the metal ball is dropped and the crater diameter formed can be expressed by the following equation.

Where k 2 and C are constants.Due to the data supporting a linear relationship, the maximum and the minimum slopes were calculated which were also used to determine the constants k 2 and C along with their respective uncertainties.

Relationship between height and crater diameter

From the graph, it can be seen that Slope: 6.109 C: 5.82 Slopemax: 6.789 Cmax: 6.00 Slopemin: 5.789 Cmin: 5.40

The uncertainty of slope k =

Or

Similarly the uncertainty of constant C is

Or

Therefore the final equation derived from this experiment is D= (6.109 0.16) h + (5.82 0.3)

Conclusion and Evaluation:-

This design lab did not confirm the original hypothesis that the height and the diameter of the crater formed are related by a power relationship. Instead it was supported by a linear relationship. The height and the diameter are linked by the equation:

D= (6.109 0.16) h + (5.82 0.3)The fact that the line of best fit passed through all the error bars and the RMSE value of 0.1681 suggested a higher reliability than that of the exponential relationship.There were several sources of error in this design lab that could have been reduced to yield even better data. First of all, despite the fact that sifter was used to extract evenly sized sand grains, during some trials, larger grains sometimes predominated the surface while sometimes this did not happen. The differences in the size of the grains ultimately affected the diameters of the crater created. This can be considered a random error because the locations of the different sized grains of sand was completely randomized and could not be controlled.Another possible error could be that the surface of the sand was not completely smoothed out between each trial, resulting in slightly uneven surfaces which affected the shape of the craters created.In addition, several trials in this experiment had to be repeated due to the diameter of some trials of some trials significantly lower than that of other. This can be attributed to the sand being compressed accidentally, resulting in a higher density of the sand, thus reducing the diameter of the crater formed.Furthermore, the method of measuring the crater diameter was flawed and unreliable. It was hard to measure the diameter of the craters with a clear ruler without disturbing the surface of the sand. Also the experiment was carried out in such a way so some sunlight was incident on the sand surface. This could have helped to identify the boundaries of the crater more clearly, but the boundaries were still unclear resulting in variable uncertainties.Two confounding variables were identified in this design lab. The first one was increased air resistance with increase with height at which the ball was dropped. Increased air resistance definitely led to loss of energy to surrounding, and less kinetic energy at the time of impact reduced the crater diameter. The second extraneous variable was the spin of the metal ball as defined earlier. The spin acquired by the ball could have affected the shape of the crater formed and consequently produced uncertainties in the diameter when measured. This factor reflects a weakness in the procedure carried out. The ball was dropped by hand at different heights, hence the initial spin could not completely controlled or held constant.

Finally, some preliminary trials could have been performed in order to estimate the optimal range of the heights chosen to be sampled. This would have prevented the need to collect to extra data points. Although the data did not confirm my hypothesis, it is justified since the hypothesis was based on an equation designed for calculating diameters of craters caused by impact with kinetic energies on an astronomical level. Thus this design lab was successful in creating an alternate model for the experiment.