Physics Lab report and the Structure, Essays (high school) of Physics

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By: Aarush Inamdar

Exploring Gravitational Acceleration

using Ping Pong Ball

1. RESEARCH AND DESIGN

1.1 AIM

Gravitational acceleration or often denoted as “g” is the acceleration experienced by an

object due to the force of gravity. It is the rate at which an object near the Earth’s surface

accelerates toward the center of the Earth under the influence of Earth’s gravitational force

2

.

This study intends to find out “ How does altering the initial height from which a ping pong ball

is dropped affect the measured gravitational acceleration, and by adding inclusion of

uncertainty in the height, time, and acceleration measurements ?” The aim of this lab report is

to investigate the effects of height on the gravitational acceleration experienced by the ping pong

ball. Understanding how the height influences acceleration can provide insights into the

relationship between gravity and the object’s position. The lab reported is grounded in the theory

of free fall, which states that all objects, regardless of their mass, fall with the same acceleration

due to gravity in the absence of significant air resistance.

1. 2 Background information

I n real life context, the air resistance will have great impact on the acceleration, as it

exerts force from the opposite direction, changing the gravitational acceleration. This

fundamental principle is described by Galileo’s experiment and the equations of motion

1 ,

. The

free fall formula can be derived from the SUVAT equation, as follows:

!

"

"

!

"

"

, 𝑎𝑠 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑠 0

"

This is the free fall formula that has been derived from a SUVAT equation, SUVAT are set

of equations used in physics to describe the motion of an object under constant acceleration of

"

. These equations are used in the kinematics

1

The S stands for Displacement (m), U for

Initial Velocity (𝑚𝑠

#!

), V for final velocity (𝑚𝑠

#!

), A as the gravitational acceleration/acceleration

( ) and T for the time taken (s).

1

The free fall formula is simplified to this due to the initial velocity

(U) typically being zero because the object is being dropped from rest. Therefore, the equation

simplifies significantly by removing the initial velocity. Through the formula the gravitational

acceleration has a direct relationship with the height (displacement) to the time of fall, which

shows that an increment in the height would fluctuate the gravitational acceleration.

Footnotes:

1. “Numeracy, Maths and Statistics - Academic Skills Kit.” Ncl.ac.uk, 2023, www.ncl.ac.uk/webtemplate/ask-assets/external/maths-

resources/mechanics/kinematics/equations-of-

motion.html#:~:text=The%20equations%20of%20motion%2C%20also,%2D%20acceleration%20and%20t%20%2D%20time. Accessed

27 Oct. 2023.

2. “The Acceleration of Gravity.” Physicsclassroom.com, 2023, www.physicsclassroom.com/class/1DKin/Lesson-5/Acceleration-of-

Gravity. Accessed 27 Oct. 2023.

Dependent Variable (Time Taken for the Ping pong ball to hit the ground)

The dependent variable in this experiment will be the time taken for the ping pong ball to

hit the ground. The reason why the time is the dependent variable is because when the ping pong

ball is dropped from a greater height, it covers a longer distance during its fall. As a result. It

requires more time to travel that increased distance. Conversely, when the ping pong ball is

dropped from a lower height, it covers a shorter distance and thus requires less time to reach the

ground. The tool that is used to measure this time is stopwatch, which is used to measure the

time taken for the ball to reach the ground. This shows that the height and the time have a linear

relationship towards each other, meaning if the height increases the time taken will also increase

and conversely. This is due to the formula 𝑔 =

"$

%

!

Controlled Variable

Variable Method Explanation

Cross

Sectional

Area

The controlled variable for this

experiment is cross-sectional area of

the ping pong ball, throughout the

experiment this variable is kept

constant to ensure that the data is

consistent and reliable to analyze.

By keeping the cross-sectional area of the ping pong ball constant,

this ensures that the air resistance experienced by the ball will

remain constant and consistent throughout the experiment

7 .

According to the hypothesis the gravitational acceleration would be

9.81𝑚/𝑠

"

without any air resistance

7

. If there were any air drag on

the ball the data would not be consistent and reliable

Air

resistance

Another controlled variable for this

experiment is the Air resistance on the

ball, throughout the experiment this

variable must be kept constant to

ensure that the gravitational

acceleration is consistent and reliable

to analyze.

As the air resistance, on the ball might slow down the acceleration

(Hypothesis)

7

. If there was any air resistance or drag, the

acceleration would not be reliable and would give a data that would

be not accurate to analyze.

Temperature Another controlled variable for this

experiment is the temperature , all

TIME, and all means the temperature

must be kept controlled, this can be

kept controlled through closing the

windows in the room.

As the wind, would affect the ball’s movement. As a high wind gaze

would make the ball fly away, and the data would show that the

acceleration is not accurate and is not reliable due to the wind. Ping

pong ball are lightweight ball which can be affected from a small

gaze of wind. If there was a high wind, the ball would move away

from the experiment, and give a wrong data.

Initial

Velocity

The fourth controlled variable for this

experiment is the initial velocity added

to the ball before leaving it. In the

experiment, there shall be no external

velocity from the experimenters

As an initial velocity would increase the gravitational acceleration,

and the ball would fall faster than gravity, creating a wrong and

insufficient data to analyze from. Making this experiment

redundant, throughout the experiment there shall be no external

velocity exerted on the ball, it must be dropped.

7. “Effect of Size on Drag.” Nasa.gov, 2021, www.grc.nasa.gov/www/k-

12/rocket/sized.html#:~:text=The%20total%20aerodynamic%20force%20is,the%20area%20doubles%20the%20drag. Accessed 27

Oct. 2023.

1.3 MATERIALS

Ping Pong Ball – The ping pong ball is the object that is being used in this experiment. The ping

pong ball serves as the object of study.

2 Rulers (1 meter) – The ruler in this experiment serves as the measurement tool, the ruler is

used to measure the height from which the ball will be dropped. In total there is a need of 2

rulers to make 2 meters.

Stopwatch – The stopwatch is used to measure the time taken for the ball to hit the ground.

Laptop – The phone is used to record the data. In total there are three people doing 5 trials for

this experiment.

Duct Tape – To make the rulers stand up tall, the duct tape will connect the two rulers and keep

them straight and high up.

1.4 PROCEDURE

Step 1 – The first step of this experiment is to gather all the materials, to conduct the

experiment itself. This includes the ping pong ball (use the Same one for the whole experiment,

due to the air resistance), stopwatch, laptop, and the ruler.

Step 2 – The second step is to set up all the materials shown in the Diagram 1.0, where the ruler

will stand up tall, in this case it would be two rulers instead of one. As there are 5 trials in total

for this experiment. There will be two rulers in total 2m, standing up tall with duct tape to

balance them.

Step 3 - After setting up all the materials, the next step is to take the ping pong ball and place it

on the right height, the first height would be 1m for this experiment.

Step 4 – The next step is to keep the stopwatch ready, as when the ping pong ball is dropped

simultaneously the stopwatch must be activated for the perfect timing and reliable data.

Step 5 – After the ball is dropped and hits the ground stop recording the time.

Step 6 – After the time has been recorded, take the laptop, and record the time for the trial.

Step 7 – Repeat this step for each trial for the initial height of 1m. In total there are 5 trials for

each height. Everyone must do this once, and an additional of 2 from 2 other.

Step 8 – Now repeat all the steps for the initial heights of 1.25m, 1,5m, 1.75m and 2m.

Step 9 – Record all the time for each trial for various initial heights.

2.1 PRESENTING RAW DATA (2 decimal places)

Initial Height

(m) ± 0. 0005 𝑚

Trial 1 (s)

Trial 2 (s)

Trial 3 (s)

Trial 4 (s)

Trial 5 (s)

Table 1 - Raw Data Table with Time and Height, with the initial stopwatch and ruler uncertainty

Initial

Height (m)

Trial 1 (s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s)

Table 2 , Gravitational Acceleration calculated for each time.

The uncertainty for the digital chronometer (phone stopwatch) is 0.001s, which means

the total uncertainty for the time would be:

𝑇𝑜𝑡𝑎𝑙 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 = 𝐷𝑖𝑔𝑖𝑡𝑎𝑙 𝐶ℎ𝑟𝑜𝑛𝑜𝑚𝑒𝑡𝑒𝑟

2.2 Equations that are being used.

Free Fall Formula:

𝑠 =

1

2

𝑔𝑡

"

𝑊ℎ𝑒𝑟𝑒 𝑠 = 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑

𝑡 = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑡𝑜 ℎ𝑖𝑡 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑

Rearranged Formula:

𝑔 =

2 𝑠

𝑡

"

Gravitational Acceleration Using Gradient:

𝑔 =

2

𝑠𝑙𝑜𝑝𝑒 𝑣𝑎𝑙𝑢𝑒 (𝑚)

This is the rearranged formula for the gravitational acceleration, where the displacement is multiplied by 2 and is

divided by time squared.

Range and Uncertainty:

𝑅𝑎𝑛𝑔𝑒 = 𝑀𝑎𝑥 − 𝑀𝑖𝑛. 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =

𝑀𝑎𝑥 − 𝑀𝑖𝑛

2

𝑜𝑟

1

2

𝑟𝑎𝑛𝑔𝑒

Percentage Uncertainty

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 % =

𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 (±)

𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

Average 𝑡𝑖𝑚𝑒

!

!

!

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑡𝑖𝑚𝑒

"

= 0. 52 ∗ 0. 52. 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑡𝑖𝑚𝑒

"

= 0. 27

!

and 𝑀𝑖𝑛

!

!

!

!

!

= 0. 46 ∗ 0. 46 𝑀𝑖𝑛

!

= 0. 21

Uncertainty in 𝑡𝑖𝑚𝑒

L

!

!

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑖𝑛 𝑡𝑖𝑚𝑒

!

=

".*𝟒&".!)

!

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑖𝑛 𝑡𝑖𝑚𝑒

!

=

".)𝟑

!

𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑖𝑛 𝑡𝑖𝑚𝑒

!

= 0. 06

2.4 Presenting Processed Data:

Initial Height

Average time, t/s Absolute uncertainty

of t/s

Average time 𝒕

𝟐

/𝒔

𝟐

Table 1 – Finding the Average time and the Absolute uncertainty.

Trial 1

#"

) (1m)

Trial

#"

(1.25m)

Trial 3(𝑚𝑠

#"

)

(1.5m)

Trial 4

#"

Trial 5

#"

) (2m)

Average

mean for

each height

Table 3 , Average of the gravitational acceleration for each height

Initial

Height

Average

time, t/s

Absolute

uncertainty of

t/s

Average

time 𝑡

"

/𝑠

"

"

"

"

"

Absolute

uncertainty of

"

/𝑠

"

Table 2 - Finding the Max and Min and Absolute Uncertainty t^2/s^

In the end the gravitational acceleration for the whole experiment is:

!

!

with the percentage uncertainty of approximately 1. 32 %

2.5 Graphs:

Graph 1 - Graph assessing the average 𝑡𝑖𝑚𝑒

! (𝑡

! /𝑠

! ) and initial height (m).

Graph 2 - Graph assessing the average time (s) and initial height (m).

2.6 Observations from graphs and data tables

The data extracted from the graph sheds light on the relationship between the initial heights

from which the ping pong ball was launched (x-axis) and the corresponding average time it took

for the ball to reach the ground (y-axis). The absence of a linear trend in the graph becomes

evident, signifying the non-linearity of the data. This non-linearity arises from the inherent

variability in the collected data, which can be attributed to factors such as reaction time and

stopwatch uncertainties. Consequently, each height value resulted in a distinct data point,

contributing to the non-linear distribution observed.

The presence of a trendline further underscores the lack of direct proportionality between the

independent variable (initial height) and the dependent variable (time taken). This nuanced

relationship implies that additional factors beyond initial height significantly influence the time

it takes for the ping pong ball to descend. As a result, the gravitational acceleration in our

experiment exhibits deviation from the established constant value of 9.81 𝑚/𝑠

"

. This deviation

is caused due to the uncertainty of the time and the stopwatch 0.005s. Additionally, graph 1

explores the relationship between the initial height (1m, 1.25m, 1.5m and 1.75m) and average

time squared. Notably, both graph 1 and graph 2 exhibit similar shapes and directional trends,

indicating that the cross-sectional area remains consistent, and the data points align.

However, an interesting observation is the downward shift of data points in graph 2. This shift is

a consequence of squaring decimal values (e.g., 0. 52

"

= 0.27), resulting in a halving of the values

and subsequently leading to the downward displacement of the corresponding data points in

The line of best line doesn’t pass through all the error bars creating a conflict between the graphs

and the data table accuracy, which means that the gravitational acceleration of 8.33 𝑚/𝑠

#"

is

not an accurate representation of the g value. From the graph, the only error bar the best fit line

passes through is the height of 1m and the value of 0.27s. The other values show that the

acceleration is not accurate representation as the line didn’t pass through them, showing that

the data representation was not the right.

3. Conclusion

3.1 Theoretical Value

The theoretical value for this experiment would be the constant gravitational acceleration

when the air resistance in negligible which is 9.81 𝑚𝑠

#"

. The theoretical value will assist to find

the percentage error between the actual value and the average value of the gravitational

acceleration calculated for this experiment. Another percentage error is the data that was

derived from the graph and theoretical value. The formula being used for this experiment is:

End Calculation for the final gravitational acceleration:

Σx

𝑛

= 𝑥̅ , 𝑤ℎ𝑒𝑟𝑒 Σx = sum of all values, n = number of values

Average for the Gravitational Acceleration:

𝑥̅ =

7. 40 𝑚/𝑠

"

• 10. 40 𝑚/𝑠

"

• 9. 57 𝑚/𝑠

"

• 10. 40 𝑚/𝑠

"

• 7. 72 𝑚/𝑠

"

5

𝑥̅ =

45. 49

5

𝑥̅ ≈ 9. 1 𝑚/𝑠

"

And Average Percentage Error Value it is:

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

|𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝐸𝑟𝑟𝑜𝑟 − 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|

𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

| 9. 1 − 9. 81 |

9. 81

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

| − 0. 7 |

9. 81

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

1. 7

2. 81

∗ 100% = 0. 071

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 ≈ 7. 13 %

And Graph Percentage Error Value it is:

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

|𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝐸𝑟𝑟𝑜𝑟 − 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|

𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

| 8. 33 − 9. 81 |

9. 81

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

| − 1. 48 |

9. 81

∗ 100%

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 =

1. 48

2. 81

∗ 100% = 0. 15

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐸𝑟𝑟𝑜𝑟 ≈ 15 %

From the percentage error both the percentages are above 5%, showing that the data is not

accurate and reliable enough. Although, the percentage error for the average value for

significantly closer to the 5% mark, but the graphing error value was 15% showing that the

whole acceleration was wrong. As a 15% percentage error is a high value compared to the 7%

from the average of the values.

3. 2 Concluding

The findings derived from this experiment underscore the challenges in obtaining accurate and

reliable data within this scientific context. The examination of the data and graphical

representations highlights discrepancies in the determination of acceleration, primarily due to

the inability of the graph to intersect with the error bars. This discrepancy underscores the

inherent inaccuracy in the calculation of the raw data, rendering it unsuitable for use as a

dependable basis for further analysis. A key concern lies in the significant percentage error,

which reaches 15%. In the realm of scientific precision, this percentage error is notably higher

than the acceptable threshold, which typically hovers around 9%. The magnitude of this error

far surpasses established limits, signifying the limitations and unreliability of the collected data.

T his distinction is a critical factor in the assessment, rendering the data insufficiently reliable

for investigating the proximity of gravitational acceleration to its expected constant value.

However, it is noteworthy that the hypothesis posited for this experiment, which suggests a

deviation in gravitational acceleration due to air resistance, was indeed substantiated.

The data collected predominantly demonstrates that the gravitational acceleration values tended

to fall below the expected constant. Nevertheless, the presence of a few outlier data points

suggests the potential influence of human error within the experimental setup. Throughout the

experiment, the graphical method and the average values of the acceleration gave a value which

was lower than the uniform acceleration which shows that the hypothesis was supported by the

data, calculations, and the graph’s slope. Contradicting, the uncertainty that was calculated using

the graphing method of Max slope – Min slope, suggests that the uncertainty level is ±0.

#"

, which is relatively very small and according to Nagwa School a smaller uncertainty level

shows the accuracy of the data. But in this experiment, the accuracy was wrong, as the highest

value possible is 8.44 𝑚𝑠

#"

and the lowest possible value is 8.22 𝑚𝑠

#"

which is not possible

because if the uncertainty were to be very close the highest and lowest must be in the levels of

#"

, but due to inaccuracy this data couldn’t be calculated.

4. Evaluation

4.1 Answering Research Question

The method of finishing the experiment for finding the gravitational acceleration helped to

answer the question, and this is due to the results, data and calculations that have been done to

see if the fluctuation in the height would change the acceleration with keeping uncertainty in

measurement. The experiment revealed a nuanced relationship between the initial height of the

ping pong ball and the measured gravitational acceleration. Contrary to the expectation of a

linear trend, the data showed non-linearity trend which didn’t start from the origin (0,0),

suggesting that additional factors beyond initial height significantly influenced the time it takes

for the ball to descend. These additional factors include air resistance, human errors, and the

influence of reaction time of a human.

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2

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3

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