Further points can then be recorded around the odd result. A new trend can now be noticed. Without plotting the extra points, the result would just have been taken as an anomaly. Therefore drawing a graph alongside the experiment place allows for greater scientific accuracy and better end results.
Graph:
In this experiment a graph of R against h1 would seem suitable at first thought.
The previously derived formula:
R=√(4x h1 x h2)
When used in a graph this formula would not give any sort of relevant and conclusive result. A straight line graph is needed to give a conclusive correlation, in the form: y=mx+c
Where:
y is the position on the y axis
x is the position on the x axis
m is the gradient
c is the point of y intercept
Therefore the formula must be rearranged in the straight line graph form.
R=√(4x h1 x h2) squared
R²=4 x h1 x h2
R²=4 x h1 x h2 +0
R²=(4h2) x h1 +0
y=mx+0
This is now in the form y=mx+c
This means a straight-line graph will be produced with R² (on the y-axis) against h1 (on the x-axis), with the gradient (m) equal to: 4h2
This will allow me to predict the shape of the graph.
Prediction:
This prediction is purely based on the formula (incomplete energy transfer from PE to KE has not been accounted for).
The height of the table is 85cm (h2)
Values of h1 from 5-45cm (in intervals of 5cm) will be used
Using the formula R²=h1 x 4h2
A graph of h1 against R² will be plotted using the above-predicted results alongside my actual results. This will enable me a direct comparison between the two.
Time plan:
- One hour will be spent performing the experiment.
- It will take 20mins to setup all the apparatus
- 30mins to take results, including disaster time (in case things go wrong)
- 10mins to pack away apparatus
Ranges:
A range of h1, from 5cm to 45cm will be recorded in 5cm intervals.
The range will be recorded from approximately 0cm-150cm
H2 will be measured before beginning the experiment, as it is a constant.
Reliability:
Unreliable results are caused by random error. When a single recording is made the result may not be the true result, it may be close, but due to experimental circumstances (for example the ball is blown slightly off course) there is an error in that particular result. Therefore repetitions of the experiment will allow these errors to be spread evenly around the correct value. Giving a true impression of the answer.
Sensitivity:
This concerns the accuracy of the equipment. In this experiment say, for example, that a microscope is used to read the values from the ruler accurately. The low power microscope allows the scientist to distinguish to an accuracy of ±0.1mm, but the ruler markings are only accurate themselves to ±1mm. This means that a compound error is formed and shows why appropriate measuring methods should be used in conjunction with measuring equipment.
In this case an attempt to improve accuracy only serves to waste time and equipment. Appropriate equipment must be applied in each situation.
Accuracy:
Accuracy is about reducing errors wherever possible to receive more accurate results.
Accuracy of measuring ruler lengths can be increased in the following ways:
When reading the ruler the eye must be at the same level as the object, otherwise an incorrect value will be seen.
When measuring an object further away from the ruler, a setsquare can be used to ensure that the object is at the same height as the reading on the ruler.
The range measuring ruler must measure from the point the ball leaves the ramp, which in this case is the edge of the table. This means that it must be accurately aligned under the table, this can be achieved using a plumb line. It is facing directly downwards, allowing the ruler to be placed directly under the point at which the ball leaves the ramp.
The original method of recording the range of the ball was to let it fall into the sand pit, this method has many draw backs. If very high levels of accuracy were required, the drop height to the precise impact in the sand would have to be measured. The sand is uneven, so this means the h2 value should be measured from the table to the very point at which the ball strikes the sand. Measuring the actual value of h2 proves to be impractical.
The range also proves to be a factor of the accuracy. The impact crater left by the ball makes measuring the range a tricky task. As the ball moves into the sand, the sand exerts a force on the ball, deforming the crater as the ball bounces away.
Another method in consideration is using blu tack. It avoids the uneven surface problem but may still have a slight impact crater problem as the ball comes in at an angle and digs into the blu tack.
It is for the above reasons that carbon paper was chosen instead of the sand pit method and blu tack method. It avoids the uneven surface and impact crater problems. The only possible problem that can occur is improper printing onto the paper, this is when the ball doesn’t strike the carbon paper with enough force so as to mark the paper underneath.
A travelling microscope could be used to measure the precise impact with the sand. But considering the level of accuracy used throughout the rest of the experiment, this would be an inefficient method.
Safety:
Safety is an important aspect of any investigation. The weight of stands, bosses and clamp combinations and “g” clamps must be considered as they may fall if knocked. The setup should be well thought through taking into account movement around the experiment.
Results:
h2 is measured as 85cm, this value remains constant throughout the experiment.
The accuracy of recorded results is dependent on the sensitivity of the equipment:
The metre rule can be read to an accuracy of ±0.5mm
With all ruler measurements in this experiment, the value above zero is measured, meaning that there is an uncertainty at either end of the ruler. Therefore giving a total uncertainty of 1.0mm or 0.1cm.
The height, h1 was measured accurately to ±0.1cm
The range was measured accurately to ±0.1cm
The height of leaving the ramp, h2, was measured accurately to ±0.1cm
These details will allow error bars to be plotted on the graph providing more a more accurate idea of how the range is affected (within the limits of experimental uncertainty).
The following graph was plotted alongside the experiment. Note points 1 and 2 identified on the graph, they were discovered as possible anomalous results. These two points were investigated by finding the range produced at nearby heights. It was concluded that the results 1 and 2 were random uncertainties.
Further plotted heights to help explain points 1 and 2:
Percentage error:
I will take the maximum height and maximum range and calculate their percentage errors based on their uncertainty values.
% error of height, h10, 45cm ±0.1cm
0.1/45= 0.0022222 *100= 0.22%
% error of the range, r, 95.5cm ±0.1cm
0.1/95.5=0.0010 *100= 0.10%
As the value is expressed on the graph as range² we must times this by 2
0.10% *2 = 0.21%
These values are so small that they are hardly worth considering when plotting a graph, which is why they have been omitted from the graphs (this is discussed in more detail later on).
Accuracy issues:
The level of accuracy of recording and also the levels of sensitivity of equipment were both appropriate to the function of this investigation. They helped to provide reasonable results with an easily identifiable conclusion. The results are reliable, although not quite as predicted the limiting factors could be identified and an accurate result attained.
The results plotted on the graph do not appear to take account for the uncertainties calculated earlier. This is because the uncertainties would have barely affected the overall outcome of the graph. Such tiny amount would be more suitable for incorporation in an extremely accurate experiment. This experiment did not require such a high level of accuracy and as such the uncertainties were not shown on the graphs.
Graph analysis:
Upon examining the graphs, a positive correlation between the squared value of the range and the height the ball is dropped from can be seen, although points vary from the line of best fit quite noticeably in some places i.e. points 1 and 2.
The results prove that the derived equation is correct and that the range of a ball does depend mostly on the height it is dropped from and the height the ball leaves the ramp. The formula (R²=4 x h1 x h2) states that an increase in the drop height, h1, increases the range of the ball. The results have proved this to be correct.
In conclusion the experiment occurs the following way:
The ball is released with high gravitational potential energy as it rolls down the slope, this energy is converted into kinetic energy, the higher the drop height, the longer the ball is travelling down the slope, so more energy can be transferred giving more kinetic energy and hence more speed when the ball leaves the ramp.
The results also show that other factors affect the range.
In the prediction it was calculated that the gradient of the graph (of h1 against range²) should be equal to 4h2, but the gradient results (from graph that compares the prediction and actual results) show that 31.4cm is lost between the predicted value of h2 and the actual value of h2. The height of the table, h2, remained constant throughout the experiment; this means that the recorded range was shorter than the predicted range. This therefore shows that ball lost speed or energy, resulting in the shorter than expected range.
The derivation is correct, but the only part that could affect the results in this way was the assumption that all energy was transferred. The transfer is likely to be less efficient than assumed, 100% energy transfer from potential energy to kinetic energy is extremely unlikely. The energy can be lost in a variety of ways which are covered below.
Limiting factors:
The level of the ground could affect the results as a whole, depending on which way the ground is sloping it could either increase or decrease the recorded range.
The above diagram is exaggerated, but a difference in the angle of the floor will cause the carbon paper to also be at an angle. In the example the recorded result will be less than it should be with a flat floor.
In future experiments a spirit level can be used to verify that the floor is flat.
Air resistance is an important limiting factor it is a form of friction that acts upon any object that moves through air. In these circumstances we can say that air has the properties of a liquid, when an object is pushed through a liquid, the liquid pushes back against the object with a force as the object moves through the liquid. The liquid is pushed to the sides of the object, this pushing requires force, so energy is lost from the ball. This means that the ball in this experiment will slow down as it uses some of its force, to push aside the air and continue moving.
Air resistance is dependant on the size of the object, as the larger the object the more air it has to push to the side. It also depends upon the speed at which the object is travelling, as an object travelling at faster speeds has to push aside more air in the same time.
This effect can be used to explain the slight curvature at the top of the graph. This is because the ball is travelling at higher speeds (because it is being dropped from greater heights), with more air resistance acting upon it due to the higher speeds, causing it to cover less distance.
Air resistance cannot be removed unless working in a vacuum, but in future experiments it would seem more sensible to incorporate a formula to account for it. Or alternatively use a denser ball, which is less affected by air resistance.
The straightness of the ramp can affect the experiment in two ways.
If the ramp is not straight the ball will bounce from side to side causing unnecessary friction and the unnecessary loss of energy.
The balls energy is lost into the side walls of the ramp, causing the ball to slow and travel a shorter distance in the air.
The ball may also travel off course, to one of the sides. This means that it covers less distance than it would have, had it gone straight down the ramp. This can be proven theoretically:
Pythagoras’ right angled triangle rule: a²+b²=c²
Both route a and route c, reach the same distance (b)
But route c covers a greater distance as a²+b²=c²
Both route c and a are travelling at the same speed and will land at the same time. As d=st, they must cover the same distance along their individual paths. This means that route c will finish at point d giving a shorter range, while route a carries on.
Therefore if the ramp is uneven and the ball doesn’t travel down the middle of the ramp, range will be lost, producing less accurate results.
In future experiments it would be sensible to use a combination of a spirit level and careful measuring to ensure the ball travels straight down the ramp. A more restricting track would be another option, it would mean the ball follows a set path.
Another factor affecting the speed of the ball is the way in which it travels down the ramp. It could slide or skid instead of rolling meaning that the ball is not gaining rotational kinetic energy and losing its potential energy as it should. Instead it is falling, resulting in an incomplete energy transfer, so the ball doesn’t reach maximum speed when leaving the ramp. This effect may be reduced using a ramp with a more gripping surface.
Even the smallest factors can contribute to an overall large effect. Sound energy caused by friction between the ramp and ball can reduce the amount of energy the ball has as it leaves the ramp. Sound energy is a factor that cannot be removed easily, unless working in a vacuum where sound waves cannot travel. But it is such a small factor that it is unlikely that it would affect the results in any considerable way.
Next time:
If this investigation were to be repeated, a different approach would be adopted. More care would be taken to ensure that the above factors were minimised. Many more results would be taken to provide a more reliable end result and fairness would be looked into seriously.
A light gate setup could be used to check the speeds at which the ball leaves the ramp, this result could then be considered with the mass and used to calculate the kinetic energy (KE=½mv²). The gravitational potential energy could also be calculated (GPE=mgΔh1) and the energy loss calculated.