G11 Physics HL
Minsuk Hong
Investigating the value of gravitational acceleration
Research Question
What is the average measured value for gravitational acceleration for velocities measured from
different displacements?
Background information
In this experiment, we are obtaining the value of gravitational acceleration using initial velocity,
final velocity and displacement. Gravitational acceleration is a constant that can be described as
acceleration received by the object due to the force of gravity acting on it. It is represented by
‘g’, and its unit is 𝑚𝑠 −2. Gravitational acceleration is a quantity of vector. That is, it has both
magnitude and direction (byjus). According to Newton’s law of gravity, the gravitational force is
the attraction forces between two bodies because they are lying in the gravitational field of
each other. The force of gravitational acceleration acts inwards and is attractive (byjus).
Theoretically, g is a constant acceleration value. So, regardless of the distance between the two
speed gates, the acceleration obtained from the experiment between the two gates should
always be constant if air resistance can be ignored. For our experiment, the distance for
measuring the difference in velocity is small, and the object falling is heavy enough that we can
ignore it and assume constant acceleration because the height difference between the object
and the earth is negligible as the displacement of the two gates is extremely small compared to
the distance between the object and the centre of mass of earth.
We will be using the SUVAT equation of
acceleration.
𝑣 2 +𝑢2
2𝑠
= 𝑎 to find a from our raw data of v, u and s to
According to the first equation of motion, 𝑣 = 𝑢 + 𝑎𝑡. By using this equation we can derive a
function which uses final velocity (v), initial velocity (u) and displacement (s) to give us the value
of acceleration (a) which is gravitational acceleration in this experiment.
First by squaring both sides in the first equation we can get the equation.
𝑟 2 = 𝑢2 + 2𝑢𝑎𝑡 + 𝑎2 ∗ 𝑡 2
By grouping 2a we can get the equation.
1
𝑟 2 = 𝑢2 + 2𝑎(𝑢𝑡 + 𝑎𝑡 2 )
2
The second equation of motion can be derived from substituting 𝑠 =
𝑢+𝑣
2
∗ 𝑡 in the first
equation of motion.
𝑢 + 𝑢 + 𝑎𝑡
∗𝑡
2
1
𝑠 = 𝑢𝑡 + 𝑎𝑡 2
2
𝑠=
1
As the second equation of motion tells us that 𝑠 = 𝑢𝑡 + 2 ∗ 𝑎𝑡 2 . We can substitute this in our
original equation which gives us.
𝑣 2 = 𝑢2 + 2𝑎𝑠
𝑣 2 − 𝑢2 = 2𝑎𝑠
𝑣 2 − 𝑢2
=𝑎
2𝑠
This is the final equation the experiment was based on in order to obtain the value for
gravitational acceleration.
Procedure
Material:
-
1 rod
-
2 clippers
-
2 vernier velocity light sensor
-
1 vernier analyzer
-
1 paper attached with weight
-
1 1m ruler
1. Position the upper velovity light gate using a clipper in a way the sensor is parallel to the
ground.
2. Position the lower velocity light gate using a holder such that it is parallel to the ground.
It must be sure that the lower velocity light gate is parallel to the upper velocity light
gate so that when an object is dropped, it does not collide with anything during its
freefall and the distance between upper gate and lower gate should be 10 cm.
3. Position another clipper above the upper velocity light gate.
4. Connect the velocity light sensor with the vernier analyzer and connect it with the
laptop.
5. Open logger pro on the laptop and access the photogates experiment.
6. Record down the distance between the two velocity light gate in meters
7. Position the paper attached with weight so that the top of the slip is at the end of the
marking holder on step 3 and drop the paper. (Be aware that the paper should not
interfere with any object in its freefall)
8. Record down the initial velocity obtained in the upper sensor and final velocity in the
lower sensor.
9. Repeat procedure 7 to 8, six times per distance between the upper sensor and the lower
sensor.
10. Repeat procedure 6 to 9 varying the distance between the two sensors each time
varying by 0.1m and record until distance reaches 0.5m. (Record 0.1m, 0.2m, 0.3m, 0.4m
and 0.5m six times)
Variable
Independent variable
·
Distance between the two sensors
·
0.1 m
·
0.2 m
·
0.3 m
·
0.4 m
·
0.5 m
Dependent variable
·
Gravitational acceleration
Control variable

Alignment of the photogates
The alignment of the upper and lower velocity light gates (photogates) is important to
measure the time accurately. Misalignment can introduce errors in the measurements.
Possible ways to ensure that the photogates are properly aligned are before each trial,
the photogates are aligned, and their positions are consistent throughout the
experiment.

Height of Release
Although theoretically, the height should not affect the final value of gravitational
acceleration, the height from which the paper is released can affect the initial velocity of
the falling object. Variations in release height can cause differences in the recorded
velocities and, subsequently, in the calculated acceleration. Possible ways to ensure the
release height is uniform: a fixed release height should be used. This can be achieved
through procedure 3, which positions an additional clip on the rod, and the sensor may
be reset every trial.

Weight of the Paper:
The mass of the paper can affect its gravitational interaction with the Earth and further
make air friction different, causing a different acceleration. Different paper weights can
lead to variations in the force of gravity experienced by the falling object. Possible ways
to uniform the weight of the paper: we can use identical paper with a consistent weight
for all trials and measure the weight of the paper before each trial to ensure uniformity.

Air current:
Air currents or drafts in the experimental area can have a significant impact on the
motion of the falling object, especially when the object used is a wide paper. The wide
surface area of the paper can make it more susceptible to the effects of air resistance
and air currents. These factors can lead to variations in the object's velocity and
acceleration measurements, potentially introducing both systematic and random errors
into the experiment. Possible ways to minimize the effect of air current is to close
windows and doors while blocking the air current generated from the air conditioner.
Recorded data
Table 1.1 data table of initial velocity, final velocity and time and the calculation results
Table 1.2 data table of Average values, percentage uncertainty and absolute uncertainty
Calculation of acceleration (gravitational acceleration)
The method I used was the equation discussed on background information
𝑣 2 − 𝑢2
|
|=𝑎
2𝑠
|
Calculation of uncertainty
0.6452 − 2.9192
| = 13.50756
2 ∗ 0.3
The method I used was using the max and min as the value for the initial velocity and final
velocity to get a range and take the uncertainty and I got rid of anomalies such as 3 rd trial of 1st
setup.
0.969 + 1.1872
− 0.969 = 0.1091
2
= absolute uncertainty
Average value is equal to the sum of all trial divided by the number of trial
0.969 + 0.993 + 1.107 + 1.1872
= 1.06405
4
To calculate the percentage uncertainty, we have to divide the absolute uncertainty with the
average value.
0.1091
∗ 100(%) = 10.2532775715% ≈ 10%
1.06405
To calculate the final percentage uncertainty for the acceleration, we have to add all the
percentage uncertainty for initial velocity, final velocity and displacement.
2 ∗ 10.2532 + 2 ∗ 8.9913 + 3.3333 = 41.822% ≈ 40%
To calculate the final absolute uncertainty, we have to times the percentage uncertainty with
the obtained value.
0.42822 ∗ 11.02001 = 4.608 ≈ 4.6
11.0𝑚𝑠 −2 ± 4.6 𝑚𝑠 −2
Data processing and analyses
Graph 1. Average difference of cubed initial velocity and final velocity (𝑚𝑠 −2 ) against
Displacement (m)
Graph 2. Gravitational acceleration (𝑚𝑠 −2 ) against Displacement (m) Graph
Although, the uncertainty of the velocity against distance was not as big, after calculation the
percent uncertainty of some data points to be over 50%. This has caused the uncertainty of
gravitational acceleration to be very high. The maximum slope and minimum slope has
fluctuated dramatically even reaching at a point where the position was different for the two
slopes. The slope uncertainty was obtained through the use of
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑙𝑜𝑝𝑒−𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑠𝑙𝑜𝑝𝑒
2
33.38 − (−27.56)
= 30.47 ≈ 3.0 ∗ 101
2
This gives us the absolute uncertainty of the slope which is ±3.0 ∗ 101 𝑚𝑠 −2.
The absolute uncertainty for the y-intercept can be obtained throught he use of
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑙𝑜𝑝𝑒 ′ 𝑠 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 −𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑠𝑙𝑜𝑝𝑒 ′ 𝑠 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
2
.
33.38 − 19.36
= 30.47 ≈ 3.0 ∗ 101
2
This gives us the absolute uncertainty of ±9𝑚𝑠 −2.
.
Conclusion
This experiment has been designed to obtain the value of gravitational acceleration. By using
the initial velocity, final velocity and distance using the equation
𝑣 2 −𝑢2
2𝑠
= 𝑎. As shown above the
value obtained for gravitational acceleration is (9.747 ± 9)𝑚𝑠 −2 . (gravitational acceleration is a
constant around 9.81𝑚𝑠 −2 ). Although, the raw data calculated showed a similar result to the
actual gravitational acceleration, the uncertainty was much higher than expected. This can
signify that the experiment was done in high precision and accuracy but has very high random
and systematic errors.
Evaluation
Systematic Errors:
1. Alignment Error:
Alignment error arises when the upper and lower velocity light gates are not perfectly
aligned. This can cause the falling object to trigger the gates at slightly different
positions, leading to inaccurate time measurements. Such misalignment can consistently
affect all trials, introducing a systematic error that skews the acceleration values. For
example, in our experiment although the light gates seemed to be aligned, after the
experiment we found out that the clip were slipping on the rod making the aligned to be
missed. To avoid this error, it has to be ensured that the photogates are aligned
carefully before each trial by checking their positions and orientations.
2. Height of Release Error:
This error is occurred due to variations in the height from which the paper is dropped.
Even though the height theoretically should not affect gravitational acceleration, it can
impact the initial velocity of the falling object. Different heights result in different initial
velocities, leading to inconsistent recorded velocities and subsequently affecting the
calculated acceleration and uncertainty as the uncertainty is obtained through the range
of the raw data divided by 2. To mitigate this error, maintain a fixed release height for
all trials by using an additional clip to position the paper consistently. This ensures that
the height variable remains constant, reducing the potential for systematic error. Also,
during the experiment there were some trials that the object was not dropped on the
same height as the clip leading to anomalies.
3. Deformation of Paper:
At the start of the experiment the paper was straight without any deformation.
However, after having lots of trials the paper ended up being crumbled and the
boundary of the paper were deformed and almost had a texture of a tissue. As the
paper were deformed the air resistance grew because the paper became tilted and
rough. To minimize this error, we could have made several identical papers with a
consistent weight for all trials and change the object thrown whenever there are
significant deformation.
Random Errors:
1. Sensor Noise:
Sensor noise represents random fluctuations in sensor readings that cannot be
eliminated. These variations can occur due to electrical interference, sensor
imperfections, or other random factors. These sensor noise can introduce minor and
unpredictable fluctuations in velocity measurements. In order to minimize the effect of
the sensor noise, we can conduct multiple measurements for each trial and calculate
the average. The statistical approach helps us to reduce the impact of random
fluctuations and provide a more reliable velocity measurement.
2. Human Measurement Errors:
Human measurement errors occur when experimenters make slight mistakes in reading
values from instruments or recording data or mistakes made within the procedure.
These errors can vary from trial to trial due to differences in experimenter judgment or
precision. To minimize the effect of human measurement errors, we can replace human
interaction to instrument detection. Additionally, conduct repeated trials and average it
to identify and correct any inconsistencies or errors in data collection introduced by
human factors.
Anomalies
There were several anomalies within the experiment for example, in trial one there is a sudden
peak in initial velocity, and it happens throughout the experiment. The anomalies caused the
average value of the data points to be off while increasing the uncertainty of the experiment.
Although it was not very high at the raw data, but after the calculation it caused the uncertainty
to be higher than 50 percent. The anomalies have possibly come from inconsistent height of the
fall of the object.
Works cited
“Gravitational Acceleration Formula - Definition, Formula and Solved Examples.” BYJUS,
byjus.com/gravitational-accelerationformula/#:~:text=Gravitational%20acceleration%20is%20described%20as.2