The simplest graph where the gradient is equal to acceleration is a velocity-time graph. I therefore must calculate time from the results by using the equation:
rearranged to:
However this formula requires the use of the average velocity throughout the whole fall, change in velocity divided by 2:
u=initial velocity
v=final velocity
u=0 as the object started from rest, so the average velocity of the whole fall is equal to v/2
The kinematic equation: s = ut+½at2 can also be used to calculate acceleration. Again, u=0 due to the fact that the object starts from rest, this simplifies the equation to:
These results comply with the AS Advancing Physics book, which suggests that the acceleration of an object in freefall is about 10m/s-2, however the actual figure for acceleration of all freefalling objects on Earth is 9.81m/s-2 so there is a small amount of error within the data, also indicated by the slight difference in the two calculated acceleration values.
I have decided to look a little more closely at the error involved in my calculations. I believe the main source of error in the actual experiment is likely to be caused by the metal strip used to break the beam of the light gate. Due to the extra 16g on one side of the block, its centre of mass has changed, possibly making it fall at a slight angle. It is likely that this effect would be exaggerated the higher the distance fallen meaning, as the metal strip is pointing increasingly further down, effectively an object of greater and greater width is breaking the beam as it goes diagonally through the strip of metal. This would result in the beam being broken for relatively longer and the measured velocity being slightly less. Also the beam is being broken when the total mass has not travelled as far, which, in relation to the very first graph would also mean the velocity could be less.
Another source of error could potentially come from the way the metal strip is attached. It is not impossible that, being attached with blu-tack, it could have moved from its perpendicular position relative to the block. If it pointed down, this could compound the above problem and decrease measured velocity. If it was pointing up, the reverse effect could be true. Of course, due to human inconsistency, the mass may not be dropped in the same way each time which could contribute further error to the data.
The initial error in measured values can be seen in the results with differing repeat values of velocity. It is these values which have been used in every calculations and errors would therefore be compounded. I have therefore added range bars to the velocity-time graph using the table below to show the maximum and minimum values for velocity recorded around the average.
The velocity-time graph has been re-drawn showing the data which was taken when the object was dropped from 0.3m and higher – i.e. the results which were established to have less error attached from the very first graph (velocity-distance fallen) when a constant acceleration could be seen. This, as shown by the equation of the line, gives a gradient and value for the acceleration due to gravity as 10.5m s-2.
The range bars have been joined to give straight lines showing the most extreme differences in gradient that the maximum and minimum recorded velocity values provide – the minimum velocity for the first piece of data has been joined to the maximum of the last (or penultimate as it was judged to give a line with a more representative gradient), and vice versa. The difference between the gradients of each of these lines is the error associated with my calculated figure of acceleration.
(4.41 – 2.26) / (0.43 – 0.24) = 11.32 m s-2
(4.2 – 2.3) / (0.43 – 0.24) = 10 m s-2
11.32 – 10 = acceleration due to gravity ±1.32m s-2.
= 10.5 ±1.32m s-2.
The same principal was applied to the distance - time2 graph. The minimum and maximum time² values were calculated by dividing the distance by the maximum and minimum average velocities of the whole fall respectively. As this error range is associated with time, the variable along the x-axis, a line of best fit through each of the maximum and minimum ranges was drawn to show the difference in gradients as before.
Gradient of average =5.1 x2 = 10.2s-2.
Extreme gradient values:
(0.178-0.057) / 0.6 = 4.96
x2 = 9.92 m s-2
(0.174-0.055) / 0.65 = 5.46
x2 10.92 m s-2
10.92 – 9.92 = acceleration due to gravity ±1 m s-2
= 10.2 ±1m s-2.
Each graph has given a slightly different value for acceleration due to gravity, and also a difference in error - rounding is probably a factor in this. Both however, are within acceptable range of 9.8ms-2 as the correct value for acceleration due to gravity lies within the ±error.
By using the actual value for acceleration due to gravity (eradicating error from previous calculations), 9.81m/s-2, I aim to prove conservation of energy by looking at the potential energy before the mass has been dropped from a certain height and the kinetic energy of the mass as it is falling. These two figures should be equal if all energy is conserved and simply transferred from potential to kinetic. The formulas below will be used:
The graph shows that both energy forms increase in direct proportion to the distance fallen. The gradients are almost exactly equal (equations of lines shown in corresponding colour) proving the conservation of energy. However there is a 0.1007 difference in gradient and the lines can just be seen to be unequal - particularly towards the lower end of the graph. I am therefore going to investigate whether there is any relationship between the distance fallen and the difference between energy forms, to see where the kinetic energy was lost – most likely as heat from friction.
There seems to be little correlation between the difference in potential and kinetic energy and the distance the object fell from. I would have perhaps expected there to be greater difference as the height increased, due to the fact that the speed of a falling object has been proved to increase with the height it has dropped from – increasing the effect of air resistance and therefore friction, creating heat. Although due to the scale of this experiment, it would be probable that any effect of this nature is very difficult to see.
In summary, I have calculated the acceleration due to gravity of a falling object and consequently recognised the error involved, before going on to prove the conservation of energy by calculating the potential and kinetic energy of the falling mass.
