Table of results (see overleaf for graphs)
Analysis of graphs
The first two graphs have produced the expected shapes save two anomalous results. They have both been highlighted on the graphs with a red circle. The first graph of Drop Height vs. Crater diameter has its anomalous result at the very lowest drop height.
This could be due to the fact that the ball might fall differently right at the start of its drop, or the fact that the ball did not have enough kinetic energy to displace the sand grains. Maybe the grains have an energy threshold, below which they do not move when struck. With more time it would have been interesting to drop more balls at these lower heights to investigate the way in which the sand behaves on impact.
Apart from this one result, the graph clearly shows a linear relationship between the height dropped and the resulting impact crater.
The second graph also seems to show a linear relationship, but once again a single anomalous result is apparent. The large depth for the drop from 50cm suggests that the sand might have been less compact than usual, as this would have meant that the ball penetrated deeper than usual, with the grains being further apart.
Experiment Two – Does the mass of the ball have an effect on the crater’s diameter/depth?
For this experiment, a special ball needed to be devised. It needed to be able to alter its mass, but keep its diameter the same. Short of painstakingly finding identically sized balls of exact masses, the best option was to cut a table tennis ball in half. The ball could then be filled with plasticine and reassembled with very thin sticky tape, so as not to alter the shape of the ball.
The ball was dropped from an ample height of 35cm with 4 different masses. Each mass was dropped 5 times and an average taken for the crater diameter and depth (to the nearest millimetre).
See overleaf for graph.
Graph Analysis
This graph is very pleasing, as both the crater diameter and depth appear to have good linear relationships with the mass of the ball.
This will be due to the ball having more kinetic energy (mass multiplied by velocity squared) – so more energy will be put into displacing the sand grains downward and outward.
Experiment Three – How does crater diameter vary with the ball’s diameter?
Balls of similar mass and varying diameters were chosen for this experiment. Again the drop height was kept constant at 35cm and the balls were dropped 3 times.
Graph Analysis
Another linear relationship – this time it is looser though. This could be due to the masses of the balls being slightly different. It was very difficult to find balls with similar masses – it would have been near impossible to find so many balls with identical masses.
Again the graph produces no surprises, as you would expect a ball with a larger diameter to produce a larger crater. The linear relationship would suggest that there is a constant that could be found and applied to balls of any diameter.
Angular Craters
Now that the basic relationships between the balls and their craters have been found by dropping vertically the next step is to let the balls enter the sand at an angle, to find more complex relationships between various components of the ball’s motion and the characteristics of the craters produced.
Equipment Setup for Angular Drops
Experiment Four – Changing the drop angle and investigating the crater’s shape.
This experiment was set up so that the ball had the same kinetic energy for each drop. The vertical height above the landing tray was kept constant so the GPE of the ball (mgh) remained the same. This meant that the entry speed of the ball would be the same at each angle. The ball was rolled 3 times and an average of crater length and depth taken.
Look at the last five points on the graph – they seem to show a type of exponential shape. The only way to know for sure is to plot a graph of log length against Angle for those last five points. The semi-log graph should be a straight line, if the relationship is exponential.
See overleaf for semi-log graph
The semi-log graph certainly shows a straight line, so the relationship between drop angles greater than 25 degrees and the crater length is exponential. However this still leaves the question of those early angles, why aren’t they following any kind of trend?
I think that the answer is due to the properties of the landing material. The sand was very good at stopping balls dropped vertically or from a steep angle, but the retardation of balls at a shallow angle was much less. Only a frictional force was available to stop the smooth ball. This meant that the ball skipped across the sand, further than expected – as shown on the graph above.
This graph shows a good linear relationship between the drop angle and the crater depth, as the vertical component of the velocity becomes greater as the angle increases.
Vertical component = a sinθ where a is the magnitude of the velocity.
Horz component = a cosθ
If a remains the same, and θ is increasing, then by using trigonometry it can be shown that sinθ increases (up to 90 degrees) whereas cosθ decreases, making the horizontal component decrease and the vertical component increase.
Experiment Five – Keep the drop angle constant and vary the vertical height.
An angle of 20 degrees was chosen for this experiment. Once again the same 16.6g ball was used.
By changing the vertical height, the ball’s energy would by affected. It would be sensible to predict that more energy would make for longer, deeper craters.
This graph is certainly unexpected. The crater length does indeed vary proportionally to the vertical height, but the crater depth appears to have no relationship whatsoever with the height (therefore the entry speed). It only appears to be affected by the entry angle, which determines the vertical component of the velocity. So this is due to the independence of horizontal and vertical motion – a law which is well demonstrated here.
Projectile Craters
The next step is to model the ball as a projectile, to investigate further the independence of horizontal and vertical components of motion.
Equipment setup
Experiment Six
This experiment was set up as shown above, and by using different combinations of plastic trays 7 different heights were tested.
NB “x” is the horizontal distance from the ramp to the centre of the point of impact.
So as the vertical height dropped increases, so does the distance travelled by the ball. This is to be expected, as the ball will have some horizontal speed from the release ramp. If the graph were continued for greater heights, I expect it would begin to level off as the drag on the ball would cause it to decelerate. Unfortunately due to the restrictions of time and the lack of height in the lab, this could not be investigated.
The next graph (overleaf) is possibly the weakest set of results. The depth once again has not been affected by the velocity of the ball, so this is encouraging. The angle of impact must be the only factor affecting the depth of a crater.
However the length of the crater did not show any strong relationship with the drop height. I would have expected the craters to get shorter as the vertical drop height increased (due the vertical component increasing), which has happened very loosely. The results for this relationship are not sound enough for a conclusion to be made.
Conclusions and Evaluation
The following relationships were found:
- For vertical drops, the drop height, ball mass and ball diameter all have a linear relationship with the resulting crater’s diameter and depth.
- The angle at which a ball enters the sand is exponentially linked with the length of the crater produced (for angles above 25 deg.).
- The angle at which a ball impacts is proportionally linked to the depth of the impact crater.
- The speed of impact has no discernible effect on the depth of the crater (at low speeds at least).
- The vertical height dropped has a linear relationship with the horz distance travelled.
On the whole, the experiment was a success, however there are a few problems which would be solved if the experiments were to be repeated.
Firstly, for the final experiment, the landing tray was not long enough to slow down balls travelling at high speed. A longer tray would have been able to produce a wider range of results.
Also the sand was difficult to keep perfectly level, and its compactness was hard to measure/control. This could have lead to some inaccurate results, as sometimes the ball would simply sink into the sand, but on other occasions it would roll across the top – giving a much larger distance. Possibly a rake or a large flat object could have been used to level the sand and break up any clumps.
