THE UNIVERSITY OF THE WEST INDIES
ST. AUGUSTINE CAMPUS
PHYS 1221, 1222 – LEVEL I LABORATORY
LAB REPORT COVER SHEET LAB 3: MOMENTUM: BALLISTICS
LAB DAY: TUESDAY
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DATE PERFORMED: ____________
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DATE DUE: ___________________
For Lab Demonstrators and Supervisors Only
MARKING SCHEME
SECTION
TOTAL
Performance
5
Abstract
4
Introduction and Theory
8
Experimental Details &
Precautions
2
Results and Calculations
16
Error Calculations
—
Discussion
11
Conclusion
2
References
2
Sub Total
50
MARKS
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Late Submission
Total
DEMONSTRATOR’S COMMENTS
50
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entered here)
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Title:
Momentum and ballistics
Objective:
This lab has been broken up into three experiments; the objects of the three labs are as
follows: to verify functional the time of flight when a projectile is launched horizontally
and to verify the predicted range of a projectile when launched at an angle when given
the initial velocity, initial vertical distance (Experiment 1). Other objects if this lab
include: determining how the range of the ball depends on the launch angle (Experiment
2) and to determine the relationship between the horizontal and vertical distances of the
projectile when it is launched horizontally (Experiment 3)
Abstract:
In experiment 1, the time of travel of the projectile was found using a simulator, and
the range of the projectile was also found when launched at an angle. In experiment 2,
the range of the projectile was found by launching at different angles at two different
elevations. In experiment three, the vertical distance of the projectile at different range
values was found and a graph was plotted. All values were obtained through the
simulator and verified through theoretical calculations. In conclusion all objectives
were achieved; In experiment 1, the time of flight was found to be 1.42 s, in experiment
1B, the time of flight was found to be 3.01s; the range of the projectile (the horizontal
distance travelled by the projectile) was found to be 49.31 m. In experiment two, it was
determined that a larger launch angle doesn’t mean a larger range; instead, it was
determined that a launch angle between 40o to 50o produces a longer projectile range
compared to greater launch angles. In experiment 3, it was determined that the
horizontal distance travelled by the projectile is independent to the vertical distance of
the projectile; this means even though the vertical distance decreases, the horizontal
distance travelled would still increase to a certain point.
Introduction:
Projectile motion can be simply described as the parabolic motion of a projectile as it
is launched horizontally or at an angle. When an object is launched horizontally, it
travels in a straight horizontal path but is then ‘bent’ downwards because of the force
of gravity. This is supported by Newton’s law of motion; an object traveling in a straight
path would continue to travel along that straight path unless acted upon by another
force; in this case the outside force acting upon the object is gravity. Gravity can be
described as the downward pulling force towards a centre of mass. The centre of mass
in this case is the earth. Gravity is not constant; it varies throughout the solar system.
On Earth, gravity has a constant of 9.8 ms-1. In this experiment the initial velocity,
initial vertical distance and launch angle was provided. The initial velocity of a
projectile (vo) is the velocity at which the motion of the projectile starts at. The initial
vertical distance of the projectile (yo) is the elevation point along the y-axis at which
the projectile is launched from. With this information, the travel time of the projectile
as well as the range was found. The range (x) is how far the projectile travelled from
its point of launch. The travel time or time of flight (t) is how long it takes for the
projectile to hit the ground; in this experiment the projectile travel time was found when
the projectile reached a point in the simulator where ‘y’ was equal to zero
Theory:
t= (2yo/g)1/2
Where t= projectile time of flight, yo= the initial vertical distance of the
projectile, g= acceleration due to gravity (9.81m/s)
𝑦 = 𝑦o + (𝑣o𝑠𝑖𝑛𝜃) 𝑡 – (1/2) 𝑔𝑡2
Where y= the point at which the projectile hits the ground (y- axis: 0m), yo= the
initial vertical distance of the projectile, vo= initial velocity of the projectile, 𝜃=
the angle at which the projectile is launched, t= the time of flight of the
projectile, g= acceleration due to gravity (9.81 m/s)
𝑥 = 𝑣o𝑐𝑜𝑠𝜃𝑡
Where x= range of the projectile, vo= initial velocity of the projectile, 𝜃= the
angle at which the projectile is launched and t= time of flight of the projectile
𝑚=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Where m= gradient of the graph, y2 and y1= to the y values of the gradient
triangle, x2 and x1= to the x values of the gradient triangle.
𝑚=
𝑔
2𝑣𝑜2
Where m= gradient of the graph, g= acceleration due to gravity, vo= initial
projectile velocity
Experimental Details:
In experiment one, the projectile motion simulator was used. Initial vertical distance of
the projectile was set to ten meters (10 m), initial velocity of the projectile was set to
twenty meters per second (20 m/s), and the launch angle was set to zero (0 degrees).
The projectile in the simulation was then launched; the simulation was stopped when
the y value was equal to zero meter (0 m). The time of flight at this point in the
simulation was then noted and recorded. The simulated value for the time of flight was
then verified by comparing it to the calculated value. Next, the initial velocity and initial
vertical distance were kept the same as before while changing the launch angle to thirtyfive degrees (35 degrees). The projectile in the simulation was then launched; the
simulation was stopped when the y value was equal to zero meters (0 m). The time of
flight and range (x value) of the projectile at this point was noted and recorded. These
values were then verified by comparing them to calculated values. In experiment two,
the initial vertical distance was set to zero meters (0 m), the initial velocity was set to
twenty- four meters per second (24 m/s) and the launch angle was set to ten degrees (10
degrees). The projectile of the simulator was then launched; the simulation was then
stopped when y was equal to zero (0 m). The horizontal distance travelled by the
projectile (range; x) was then recorded. This procedure was repeated for the launch
angles: 20, 30, 40, 50, 60, 70 and 80. All results were recorded and tabulated. Next,
experiment two was repeated; this time the value of the initial vertical distance was set
to twenty meters (20 m). In experiment three, the initial vertical distance was set to
forty meters (40 m), initial velocity was set to forty- five meters (45 m) and the launch
angle was set to zero degrees (0 degrees). The projectile in the simulation was then
launched; the simulation was then paused when the horizontal distance travelled by the
projectile was equal to one hundred and twenty meters (120 m). At this point, the
vertical distance (y) of the projectile was noted and recorded. The procedure was then
repeated for readings of the vertical distance of the projectile at horizontal distance
points of: (110, 100, 90, 80, 70, 60, 50, 40) m
Precautions:
The y value to used to stop the simulation to take readings was kept as close to zero as
possible
Results: EXPERIMENT 1A
Table 1A: Verifying the Time of Flight when a Projectile is Launched Horizontally
Initial Vertical distance (yo/ m)
Initial velocity (vo/ ms-1)
Launch Angle (𝜃/o)
Calculated time of flight (t/ s)
Simulated time of flight (t/ s)
10
20
0
1.43
1.42
Results: EXPERIMENT 1B
Table 1.2: Predicting the Range of a Projectile Launched at an Angle
Initial Vertical distance (yo/ m)
Initial velocity (vo/ ms-1)
Launch Angle (𝜃/o)
Calculated time of flight at an angle (t/ s)
Predicted range at an angle (x/ m)
Simulated time of flight at an angle (t/ s)
Simulated range at an angle (x/ m)
10
20
35
3.02
49.47
3.01
49.31
Results: EXPERIMENT 2A
Table 2.1 Shooting to a Level Surface
Launch Angle (𝜃/o)
10
20
30
40
50
60
70
80
Horizontal Distance (Range) (x/m)
20.09
37.66
50.71
57.73
57.85
50.88
37.76
20.69
Results: EXPERIMENT 2B
Table 2.2 Shooting from an Elevation
Launch Angle (𝜃/o)
10
20
30
40
50
60
70
80
Horizontal Distance (Range) (x/m)
58.85
68.11
74.41
75.93
71.43
60.60
44.00
23.13
Results: EXPERIMENT 3
Table 3.1: Data Table of Horizontal and Vertical Distances
Horizontal Distance
(x) (x/m)
Projectile Vertical
Distance (y) (y/m)
120
5.07
110
10.59
100
15.41
90
20.40
80
24.47
70
28.08
60
31.20
50
33.85
40
36.12
Initial Vertical Distance, yo, (Height) = 40 m
Initial Vertical
Distance – Projectile
Vertical Distance y’ =
(yo – y)/ m
34.93
29.41
24.59
19.60
15.53
11.92
9.80
6.15
3.88
Results: EXPERIMENT 3
Table 3.2: Data Table
Item
Slope of Graph (m/ m-1)
Initial speed from Slope (ms-1)
Initial speed from simulator (ms-1)
Value
0.0024
45.2
45
x2 (x2/m2)
14400
12100
10000
8100
6400
4900
3600
2500
1600
Calculations: Experiment 1A:
t= (2yo/g)1/2
t= (2[10]/ [9.8])1/2
t= 1.43 s
Calculations: Experiment 1B:
𝑦 = 𝑦o + (𝑣o𝑠𝑖𝑛𝜃) 𝑡 – (1/2) 𝑔𝑡2
𝑦 = 10 + (20𝑠𝑖𝑛35) 𝑡 – (1/2) (9.8) 𝑡2
𝑦 = 10 + (11.47) 𝑡 – (4.9) 𝑡2
Since it is in the form of a quadratic equation
𝑡=
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
𝑡=
−(11.47) ± √(11.47)2 − 4(−4.9)(10)
2(−4.9)
t= 3.02 s
𝑥 = 𝑣o𝑐𝑜𝑠𝜃𝑡
𝑥 = (20) (𝑐𝑜𝑠35) (3.02)
𝑥 = 49.47 m
Calculations: Experiment 3: (gradient of graph)
𝑚=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑚=
36.6 − 2.3
15100 − 1000
𝑚 = 0.0024 𝑚−1 (𝑠𝑓)
Calculations: Experiment 3: (initial velocity based on slope of graph)
𝑚=
𝑔
2𝑣𝑜2
Make vo the subject of the formula
vo= (g/ 2m)1/2
vo= (9.8/ 2 [0.0024])1/2
vo= 45.2 ms-1 (1dp)
Discussion:
In experiment 1A, the time of flight was found to be 1.42 s by using the simulator, it
was then verified by a calculated value of 1.43 s. In experiment 1B, the time of flight
was found to be 3.01s by using the simulator, it was then verified by a calculated value
of 3.02 s; the range of the projectile (the horizontal distance travelled by the projectile)
was found to be 49.31 m by using the simulator, which was then verified by a calculated
value of 49.47 m. These values obtained in experiment 1A and 1B were very close to
the calculated or theoretical values and therefore thought to be accurate; the minor
difference between the two values can be accounted for by the limitation of the
simulator. The simulator could not stop at exactly the point where y was equal to zero
(y being the vertical distance of the projectile); but was kept as close to zero as possible
to give the most accurate reading possible. In experiment 2A and 2B, graphs were
plotted and was found that the maximum angle to achieve the furthest projectile range
were 50o and 40o respectively. When shooting from a higher elevation, the projectile
range is greater. With a higher elevation a smaller angle is required to achieve
maximum projectile range compared to a lower elevation which requires a larger angle
in order to achieve maximum projectile range. In experiment 3, a graph with a best fit
straight line was plotted and the slope of the graph was calculated; this calculated value
helped to verify the accuracy of the values obtained by further calculating the initial
velocity of the projectile. The calculated initial velocity was found to be 45.2 ms -1
which is very close to the 45 ms-1 that was input into the simulator; therefore, all values
are thought to be accurate. The graph itself indicates a direct relationship between y’
and x2; which means that as x2 increases then y’ will also increase.
Limitations/ Difficulties:

The simulator could not be stopped or adjusted to where y was exactly to zero
Conclusion:
In conclusion all objectives were achieved; In experiment 1, the time of flight was found
to be 1.42 s, in experiment 1B, the time of flight was found to be 3.01s; the range of the
projectile (the horizontal distance travelled by the projectile) was found to be 49.31 m.
In experiment two, it was determined that a larger launch angle doesn’t mean a larger
range; instead, it was determined that a launch angle between 40 o to 50o produces a
longer projectile range compared to greater launch angles. In experiment 3, it was
determined that the horizontal distance travelled by the projectile is independent to the
vertical distance of the projectile; this means even though the vertical distance
decreases, the horizontal distance travelled would still increase to a certain point.