AUSTRALIAN INTERNATIONAL SCHOOL
SHARJAH, U.A.E
Physics Lab Report
Laws of Motion
Name:
Date:
Teacher:
Title:
Neil Joseph
2/01/2014.
Mr.Balajee.B.C
Investigation on Projectile equations of Motion.
Research question:
Can gravity be determined from the equations of projectile motion using practical
applications?
Aim:
To investigate on newton’s law of projectile motion.
Introduction:
I would like to investigate the effect of the change in the angle, pull force on the range
of projectile motion. I am going to prove this by obtaining the value of gravity from
these equations.
Abstract:
In this experiment I’m using both trigonometric and practical concepts to prove the
relationship between projectile equations and gravity I am using equations to justify my
hypothesis and my concept. I have used a lab setup or in other words an apparatus to
prove the credibility of these equations of motion. I used paint to determine the
distance or in other words the displacement of the object after impacting the ground.
Distance (range) = ∆D= [Initial position-Final position].
Hypothesis:
Can we determine the relationship of projectile equations and gravity?
Yes we can, there are many projectile equations of motion, which can help in proving
the value of existing gravity:
The equation:
R=
𝑼𝟐 𝑺𝒊𝒏𝟐𝜽
𝒈
Where “T” is time in seconds and “u” is velocity in meters per second. ”g” is gravity. And
“H” is height in meters; “R” is range in meters.
Background information:
This design can be used to determine the gravitational value from the projectile
equations of motion.
Components in the apparatus:
Horizontal component:
X=𝑢2 velocity (u) in meters per second (m/s).
Vertical component:
Y=R Range (R) in meters (m).
Background knowledge/information:
Deriving formula to determine Range using gravityDistance (S) =velocity (v)* time (t).
Hence when applying the concept of vector resolution we determine that velocity (v) is
a horizontal component and can be further quantified as,
Velocity (v) = u. Cosθ.
𝟐𝒖.𝑺𝒊𝒏𝜽
And time (t) =
.
𝒈
And since Distance (S) = Range (R)
[Projectile motion equation].
We then obtain an equation- Range (R) =
𝐮.𝐂𝐨𝐬𝛉∗𝟐𝒖.𝑺𝒊𝒏𝜽
𝒈
.
Therefore,
R=
𝑼𝟐 𝑺𝒊𝒏𝟐𝜽
.
𝒈
Where “R” is range in meters, ”U” is launch velocity in meters per second, “g” is gravity
in meters per second squared and 𝜽 angle of projectile launch in degrees.
Therefore to graph this motion𝑼𝟐 𝑺𝒊𝒏𝟐𝜽
R=
𝒈
Y=MX+C (Equation of straight line).
Therefore by relating both equations we get,
𝒖𝟐
Y-axis = range (R).
X-axis = 𝑺𝒊𝒏𝟐𝜽.
Slope (m) = 𝒈 .
Variables:
The variables here are force, mass and angle.
NAME
TYPE
HOW IS IT MEASURED?
IS IT CONTROLLED?
−1
Velocity
Independent
Meters/second (m 𝑠 )
Yes
Mass
Dependent
Kilograms (kg)
Fixed
Angle 𝜽
Independent
Degree (𝜃˚)
Constant
Apparatus (equipment):
1) A table tennis ball.
2) Paint of any dark color.
3) A protractor.
4) Measuring tape.
5) Tissues or paper.
6) Elastic rubber band.
7) Inclined plane (adjustable to any angle).
8) White board erasable marker.
Basic Preparation of apparatus:
1) First prepare an adjustable board that can change its angle.
2) Now prepare a launch stem or in other words a slingshot using the rubber band.
3) Make sure that the stems holding the slingshot is firm.
4) Now draw down a small scaling on the board using a scale.
5) After the launch board is designed take the table tennis ball and pull it against
the rubber band and launch it.
6) Once the ball has impacted the ground make sure you record the point or
location.
7) Now measure the distance from the top of the launch board to the impact point.
8) Before you take the readings make sure that you have an organized table to
record your readings on different criteria’s such as angle of launch, pull measure
and impact distance.
9) Record and label you readings.
Precautions to be taken:
1) Ensure that the inclined plane has no systematic error.
2) Take precaution in making sure that the protractor used does not present any
calibration errors.
3) Make sure that the rubber band used does not have any tear defects.
4) Do not use different rubber bands on different readings.
5) In order to ensure that velocity is constant to obtain required readings make
sure that the ball is pulled to same length in all cases.
6) Ensure that the inclined plane is always pointed to one direction in order to
eradicate any systematic errors.
7) Make sure that the apparatus is not exposed to an open environment, which
could disrupt the motion of the ball e.g.: air current.
Raw data:
Three trials of range readings where obtained for each angle ranging from (20°- 45°).
Angle θ ± 1°
20°
25°
30°
35°
40°
45°
Range R /m± 0.1m
Trial 1
Trial 2
Trial 3
1.3
1.4
1.3
1.9
1.9
2.0
2.5
2.4
2.4
3.6
3.4
3.2
3.7
3.9
4.1
4.2
4.5
4.3
Processed data:
Angleθ± 1°
Average Range R/m
20°
25°
30°
35°
40°
45°
1.3
1.9
2.4
3.4
3.9
4.3
Range m
Maximum value Minimum value
1.4
1.3
2.0
1.9
2.5
2.4
3.6
3.2
4.1
3.7
4.5
4.2
Uncertainty ± R
Error % ± R
0.050
0.050
0.050
0.200
0.200
0.075
3.8
2.6
2.0
5.8
5.1
1.7
Sample Calculations:
Results can be observed in the processed table above.
1) Average Range R/m =
𝑻𝒓𝒊𝒂𝒍 𝟏+𝑻𝒓𝒊𝒂𝒍 𝟐+𝑻𝒓𝒊𝒂𝒍 𝟑
.
𝟑
E.g.:
2) Uncertainty ± R =
1.3+1.4+1.3
3
= 1.3.
𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒗𝒂𝒍𝒖𝒆−𝒎𝒊𝒏𝒊𝒎𝒖𝒎 𝒗𝒂𝒍𝒖𝒆
.
𝟐
E.g.:
3) Uncertainty % ± R =
𝐔𝐧𝐜𝐞𝐫𝐭𝐚𝐢𝐧𝐭𝐲 ± 𝐑
𝐀𝐯𝐞𝐫𝐚𝐠𝐞 𝐑𝐚𝐧𝐠𝐞 𝐑/𝐦
1.4−1.3
2
= 0.05
*100.
E.g.:
0.05
1.3
*100 = 3.8
Graphical representation:
The two quantitates to represent newton’s law of motion are velocity and range.
There are two separate graphs designed to show the motion of the projectile in
different angles.
X-axis (sin2θ)
0.642
0.766
0.866
0.939
0.984
1
Y-axis
(Range)
1.3
1.9
2.4
3.4
3.9
4.3
Range ± 0.1 m
Max
Min
1.4
1.2
.
.
.
.
.
.
.
.
4.2
4.4
Graphs:
y = 8.2203x - 4.2535
Sin2θ vs Range
5
4.5
Range R/m ± 0.1 m
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
Sin.2θ
0.8
1
1.2
Equation- Y =8.2203 x-4.2535
Max-Min graph
5
y = 8.9385x - 4.5385
4.5
Range R/m ± 0.1 m
4
3.5
3
2.5
2
1.5
1
y = 7.8212x - 3.6212
0.5
0
0
0.2
0.4
0.6
Sin.2θ
0.8
1
1.2
Equation of Max. Slope – Y = 8.9385 x-4.5385
Equation of Min. Slope – Y = 7.8212 x-3.6212
Maximum gradient
Minimum gradient
8.9385
7.8212
Calculations to obtain gravity from graph:
Obtained slope valueSlope (m) = 8.2203
Determining velocityUsing,
𝒖𝟐
(Gravity) g =
𝒎
After transposing we get,
𝒖𝟐 = 𝒈 ∗ 𝒎.
U=√𝒈 ∗ 𝒎
−𝟏
Therefore assuming gravity is 9.81 m𝒔 .
U=√𝟗. 𝟖𝟏 ∗ 𝟖. 𝟐𝟐𝟎𝟑
Since velocity was recorded as 8.98 m 𝒔−𝟏 .
Therefore when substituting velocity𝒖𝟐
(Gravity) g =
𝒎
8.982
g =
= 9.80
8.2203
Calculating uncertainty in slope (m)Max. Slope = 8.9385
Min. Slope = 7.8212
8.982
g. Max =
= 9.02
8.9385
8.982
g. Min =
= 10.3
7.8212
Uncertainty =
𝐌𝐚𝐱.𝐠𝐫𝐚𝐯𝐢𝐭𝐲−𝐌𝐢𝐧.𝐠𝐫𝐚𝐯𝐢𝐭𝐲
=
𝟐
10.3−9.02
2
=
1.28
2
= ± 0.64
Since,
g=
8.982
8.2203
Therefore my obtained gravity value is
= 9.80 m𝒔−𝟏
9.80± 0.64 m𝒔−𝟏 .
Conclusion:
I have concluded that newton’s projectile laws of motion are applicable to practical
values. I have obtained a gravity value of 9.80± 0.64 m𝒔−𝟏 , which is close to the actual
value of gravity (9.81 m𝒔−𝟏 ).
My graph presents a linearized line in which the independent variable Sin2𝜽 ± 1˚
increases as the dependent variable Range R/m increases.
My errors also are reasonable not accurate since there was a lot of systematic error,
which can be observed in the graph (graph not originating from (0,0), and also a lot of
human error was involved when it came to observing distance of projectile impact.
My graph also shows a presence of anomalies and some error bars are very minute.
I have not mentioned any error bars since the angle was calculated using technology
and therefore is considered accurate, which explains the absence of error bars for angle
(x-axis).
Evaluation:
There were a lot of human errors involved as I obtained a 5.8 % uncertainty in my
readings I believe by recording more readings of my apparatus I could have reduced the
level of human errors in my apparatus. I had identified a systematic error, I believe my
projectile was not necessarily launched at the angle I intended for it to be launched.
Also my graph does not originate from the origin, which strongly shows a presence of
systematic error. The location where I performed or operated my apparatus was an
isolated hallway and seemed to be a perfect environment to proceed with my
experiment.
My projectile was also presenting some human error i.e. the launched projectile (ball)
was not always moving in the same direction as the other readings did.
My inclined plane was also not strong or rigid so keeping the angle steady was a
challenge. Though the inclined plane presented a marker, which pin points the angle on
the protractor I believe that the marker is not correct and it proved to show a
calibration errors. I have obtained gravity as 9.80± 0.64 m𝒔−𝟏 which is close to the
actually value but since the value is not exact I can say there is a presence of error
values.
I believe I could have improved in the field of recording the position of impact as i was
performing. I used my eyes to track the ball when it launched and when it impacted so I
believe there might be parallax error of where I observed the ball impacting and
therefore might have recorded inaccurate readings therefore bring uncertainties in the
processed data.
The inclined plane was not strong it was rather lose so a slight touch could change the
angle so I believe my readings could get accurate if my inclined plane has a rigid screw
to lock it at the angle I intend to keep it.
And I did not use the same table tennis ball for all my readings.
I also believe I could have used the help of another student to record the point of
impact. The operation of the projectile operation has taken up 3 lab sessions close to 3
hours.
Bibliography:
http://calculator.tutorvista.com/projectile-motion-calculator.html
(Helps in obtaining unknown values by transposing online in order to counter check with
obtained practical values).
http://formulas.tutorvista.com/physics/projectile-motion-formula.html
(Provides more brief info about the application and concepts of the equations of
projectile motion).