Physics IA Lab Report
Research question
:
Study of Projectile Range Vs Launch Angle
Aim
:
To find out the relationship between projectile range
and launch angle and thus find the launch velocity.
Variables
:
Dependent
:
Projectile Range (R)
Independent
:
Angle of projection (θ)
Constant
:
Speed of projectile motion (v);
Room temperature (0C)
Apparatus
:
protractor; metre rule; elastic band; test-tube holder;
blue tack
Uncertainty
:
Data collection
:
Apparatus
Least Count
Uncertainty
Protractor
10
±0.50
Metre rule
0.1 cm
±0.05 cm
I.
Constant
II.
Data Table
-
Range/cm
Angle/0
(±0.50)
:
(±0.05 cm)
R1
R2
R3
R4
R5
15
22.5
23.5
22.0
23.0
25.0
25
34.0
32.0
31.0
38.0
35.0
35
44.5
46.0
45.0
45.5
47.5
45
69.0
63.0
68.0
65.0
64.0
55
44.0
49.0
42.5
46.0
44.0
65
34.0
33.0
39.0
33.0
36.0
75
23.5
22.0
24.0
21.0
27.2
85
14.0
11.0
17.0
14.5
15.0
Albert Tanoto – 5 CKS
Formula Involved
𝑅=
𝑢2 .𝑠𝑖𝑛2θ
𝑔
:
;
Average range =
𝑅1+𝑅2+𝑅3+𝑅4+𝑅5
5
;
Uncertainty of displacement ΔD =
Rmax−Rmin
2
;
𝑅
cm  m = 1000
Data collection and processing :
Range/m
(±0.0005 m)
Angle/
(0.50)
15
R1
0.0225
R2
0.0235
R3
0.0220
R4
0.0230
R5
0.0250
Average
Range/m
0.0232
ΔR/m
0.0015
25
0.0340
0.0320
0.0310
0.0380
0.0350
0.0340
0.0035
35
0.0445
0.0460
0.0450
0.0455
0.0475
0.0457
0.0015
45
0.0690
0.0630
0.0680
0.0650
0.0640
0.0658
0.0030
55
0.0440
0.0490
0.0425
0.0460
0.0440
0.0451
0.0033
65
0.0340
0.0330
0.0390
0.0330
0.0360
0.0350
0.0030
75
0.0235
0.0220
0.0240
0.0210
0.0260
0.0233
0.0025
85
0.0140
0.0110
0.0170
0.0145
0.0150
0.0143
0.0030
Angle/
Sin 2θ
Average Range/m
15
25
35
45
55
65
75
85
0.50
0.77
0.94
1.00
0.94
0.77
0.50
0.17
0.0232
0.0340
0.0457
0.0658
0.0451
0.0350
0.0233
0.0143
#
Albert Tanoto – 5 CKS
Theory
𝑅=
:
𝑢2 .𝑠𝑖𝑛2θ
𝑔
Where: 𝑅 = 𝑟𝑎𝑛𝑔𝑒, u = initial velocity, g = gravity
At R, the vertical displacement is zero; therefore, we can find
time from the equation 𝑠 = 𝑢𝑡 −
1
𝑔𝑡 2 ,
2
By substituting t to equation 𝑠 = 𝑢𝑡 −
which is 𝑡 =
1
𝑔𝑡 2
2
2𝑢.𝑠𝑖𝑛𝜃
.
𝑔
to find horizontal
displacement, we can get R.
Method
:
1. Stick a protractor onto the edge of the table with a blu-tack.
2. Stick a test-tube holder next to the protractor with blu-tack.
3. Launch the elastic band using the back of the test-tube
holder by hooking one end of the elastic band and holding the
other with your thumb.
4. Make sure you keep the initial velocity of the launch
constant by stretching the elastic band to the same length all
the time.
5. Tabulate your data. (8 angle readings, 5 trials each)
6. Enter the results into spreadsheet and plot graphs of range
vs. angle.
Calculation (Sample)
:
1.
Sin 2(15) = sin 30 = 0.5
2.
Avg Range =
0.0225+0.0235+0.0220+0.0230+0.0250
5
Avg Range = 0.0232 mm
3.
ΔD =
0.0250−0.0230
2
= 0.0015 m
*The data used is based on the first trial of the experiments
Albert Tanoto – 5 CKS
Graph:
Projectile Range Vs Angle of Projection
Range/m
0.0700
0.0600
0.0500
0.0400
0.0300
0.0200
0.0100
0.0000
Angle/0
0
10
20
Graph 1. Projectile Range Vs Angle of Projection
Albert Tanoto – 5 CKS
30
40
50
60
70
80
90
Sin 2θ Vs Average Range
Average Range/cm
70
(1, 65.8)
60
Sin 2θ Vs Average Range
50
Anomalous
y = 42.587x + 4.0854
40
Linear (Sin 2θ Vs
Average Range)
30
y = 39.474x + 5.938
Linear (Max)
y = 36.36x + 7.7906
20
Linear (Min)
10
0
Sin 2θ
0
0.2
Graph 2. Graph of sin2θ Vs Average Range
Albert Tanoto – 5 CKS
0.4
0.6
0.8
1
1.2
Conclusion:
 There is a systematic error in the experiment because the linear line (sin 2θ vs average range) in
graph 2 does not intercept the origin.
When x = 0, y = 39.474(0) + 5.938
y = 5.938
 There are also random errors in the experiment because of the presence of the error bars,
ranging from 0.0015 to 0.0030.
 From graph 2, we can find the velocities of elastic band.
𝑅=
𝑣2
𝑔
𝑠𝑖𝑛2𝜃 + 4.2507,
Taking g = 9.81ms-2,
𝑣2
9.81
here we take R = y;
𝑣2
𝑔
= m; 𝑠𝑖𝑛2𝜃 = x; 4.2507 = c
= 39.474
∴ v = 19.7 cms-1 = 0.197 ms-1
𝑣2
Maximum velocity  9.81 = 42.587
∴ v = 20.4 cms-1 = 0.204 ms-1
Minimum velocity 
𝑣2
9.81
= 36.36
∴ v = 18.9 cms-1 = 0.189 ms-1
 Fractional error
-From the experiment (graph), ∆𝑉 =
∆𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
=
0.0075
0.197
𝑉𝑚𝑎𝑥−𝑉𝑚𝑖𝑛
2
=
0.0204−0.0189
2
= 0.0075 ms-1
= 𝟎. 𝟎𝟑𝟖
-Using apparatus uncertainties, by taking log and differentiating the equation 𝑉 2 =
2∆𝑉
𝑉
=
∆R
𝑅
+
∆𝑠𝑖𝑛2𝜃
𝑠𝑖𝑛2𝜃
, where ΔR = uncertainty of meter rule, R = average range and
Δsin2θ = uncertainty of protractor
Using the data from θ = 450,
Albert Tanoto – 5 CKS
2∆𝑉
0.197
=
0.0005
0.017
+ 1
0.658
= 𝟎. 𝟎𝟎𝟐
𝑅𝑔
:
𝑠𝑖𝑛2𝜃
Evaluation:
 The fractional error from the experiment is 0.038 whereas the fractional error from the
apparatus is 0.002. This means that there are some systematic and random errors in this
experiment
 Systematic errors
Some probable causes of the systematic errors are firstly due to the set up of the apparatus. The
protractor might not be perpendicular to the table and so the angle measure is wrong throughout
the experiment. The test tube holder (the launching stick) has a thickness of about 1 cm and so it
was not aligned properly with the protractor. This make the range measured also wrong
throughout the experiment. The systematic error could be minimize by experimenting using a
thinner stick (a wooden ruler) and by making sure the protractor is perpendicular to the table
using a triangle ruler. Secondly, there are presence of air resistance in the experiment, so our
range is lower. Thirdly, the rubber band was projected slightly beneath the angle measured
because of the arrangement of the rubber to the test tube holder. This error again can be
minimize by using a thinner stick.
 Random errors
There are some probable causes for the random errors. Firstly, it is because of the uncertainties
of the apparatus. The range measured could be higher or lower than the actual. Secondly, when
doing the experiment, the band was sometimes tilted, causing slightly more tension and
therefore higher velocity. Hence, the ranges vary. Thirdly, our hand is always shaking and could
not hold the stick still; this results in inconsistent ranges as well. Fourthly, the temperature of the
room might expand the rubber band slightly so the tension might decrease and hence again the
ranges vary. These can be improved by aligning the band correctly before launching it so it is not
tilted, keeping the room constant at all time, and repeating the experiments several times.
 Average velocity of launch (v) = 0.197 ± 0.008 ms-1
Albert Tanoto – 5 CKS