NAME
CLASS
Lab 4: Projectile Motion
Date due:
Date performed:
Objectives:
The purpose of this lab was three fold. First, to determine the initial velocity of a projectile when
θ is equal to zero. Second, to use the launch velocity when θ is zero to determine,
mathematically, the range of the projectile at a different angle. Finally, to examine the
relationship between the range of a projectile and the projectile’s launch angle.
Introduction:
This experiment is an application of the kinematic equations. The kinematic equations describe
objects in free fall, which the projectile by definition is. Once an object is under only the force of
gravity, it is said to be in free fall. A projectile is simply an object in freefall with a constant,
nonzero, horizontal velocity. Knowing that only gravity is accelerating the object, one can
approximate where the projectile will land using the kinematic equations. There is some error in
the calculation of range, as described in the conclusion. The experiment is an interplay of data
collection, reduction, and application; as explained in the methodology section.
Materials:
The object observed was a metal sphere. The sphere was placed in a spring loaded launcher. The
spring loaded launcher is what gives the ball its initial velocity, before it is in free fall. The
launcher has a plumb bob that is set against a graduated scale of degree measures. The degrees
are listed to the nearest tenth of a degree. The angle of the launcher, relative to the Earth, can be
known using the plumb bob. The launcher has three settings of force. The settings are short,
medium, and long. These settings correspond to the compression of the spring inside. For the
experiment the medium setting was used consistently. To identify where the projectile lands a
piece of printer paper over cardboard was used. The projectile leaves an indentation on the paper
when it strikes. Microsoft Excel was used for data reduction and to generate a plot. Finally, a
meter stick is used to determine the distance from a point, directly below the barrel and coplanar
with the cardboard, to the depression in the paper. The meter stick is also used to determine the
vertical distance from the barrel of the launcher to the floor. The meter stick measures to one
millimeter.
There are multiple equations used in the data reduction. They are all derivations of the kinematic
equations. All equations are using a coordinate system where up is the positive y direction and
right is the positive x direction. The first equation determines the vertical and horizontal
components of velocity:
Eq 1) V0x=V0cos θ0
and
V0y=V0sin θ0
Since the only acceleration present is that due to gravity, the following equations for the
components of acceleration are used:
Eq 2) ax=0 and
ay=-g=-9.80 m/s2
The equations used to determine the position of the projectile at any time (t) were:
Eq 3) X=V0xt
1
Eq 4) y=y0+V0yt+2ayt2
When the initial velocity is known the following equation is used to determine the time (t) the
projectile hit the ground:
Eq 5a)
−𝑉0𝑦 ± √𝑉0𝑦 2 − 2𝑎𝑦 𝑦0
−𝑉0 sin 𝜃0 ± √(𝑉0 sin 𝜃0 )2 − 2𝑎𝑦 𝑦0
𝑎𝑦
=
𝑎𝑦
To find the range (R), when the range is equal to the x value if y=0 the following equation is
used:
6a) R=(V0cos 𝜃0)t
Equations 5a and 6a are simplified if 𝜃0=0. When the projectile is launched horizontally, the
following equations can be used:
5b) t=
±√−2𝑎𝑦 𝑦0
𝑎𝑦
−2𝑦0
=√
𝑎𝑦
6b) Range=V0xt or V0x=
with ay=-g=-9.8m/s2 and assuming t> 0
𝑅𝑎𝑛𝑔𝑒
𝑡
Method:
Part I:
The magnitude of the projectile was determined by firing the projectile five times. The launcher
was in a horizontal position (𝜃0 = 0) when firing. The distance from the point of impact to a
point coplanar and directly under the barrels opening was measured. The distances were
averaged, giving an average range (x). The change in y was measured by measuring from the
bottom of the barrel to the floor. This is where the origin of any graph or equation is located in
this experiment.
Data Set 1:
Y0=10 cm X0=0
Trial results:
X=138.7cm, 141.4 cm, 140 cm, 139.2 cm, 137.7 cm,
138.7+141.4+140+139.2+137.7
Average X=
5
Average X=139.4 cm
The initial launch velocity (V0) was determined using equation 5b and then plugging the
resultant time into equation 6b. Equation 6b gave the initial velocity.
−2.1022 𝑚
t=√ −9.8 𝑚/𝑠
t=.4567 seconds
1.394 𝑚
V0= .4567𝑠
V0=3.05233 m/s
The velocity of the projectile out of the barrel was determined to be 3.05233 seconds.
Part II:
The barrel was tilted to forty five degrees. The vertical height to the bottom of the barrel was
measured. The theoretical range was computed using equations 6a and 5a. These two were used
as 𝜃 ≠ 0. Equation 1 had to be used to find the V0y before equations 5a and 6a could be used.
Data Set 2:
𝜃 = 45 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
Y0=1.039 m
V0x=V0cos θ0=V0x=3.05233cos 45
=2.15832 m/s
V0y=V0sin θ0=3.05233sin45
=2.15832 m/s
1
y=y0+V0yt+2ayt2
1
0=1.039m+(2.15832m/s)t+2-9.8t2
t=.73 seconds
Range=V0xt
=(2.15832m/s)(.73s)
X=1.5755736 m
The projectile was fired under the above conditions (data set 2). On a piece of paper a line was
drawn at 1.5655736 meters from the point directly below the opening of the barrel. The
projectile was fired and landed close to the line. This was repeated four more times and the data
and its average are below.
X=1.624 m, 1.644 m, 1.633 m, 1.686 m, 1.538 m
Xavg=
1.624+1.644+1.633+1.686+1.538
5
Xavg = 1.625 meters
the theoretical X and that difference is divided by the actual. The result is multiplied by 100 to
get a percent.
The percent error was calculated. This error is caused by reasons explained in the conclusion.
The Xavg is subtracted from
1.625 − 1.575576
(100)
1.626
3% error
Part III
Excel was used to record data from several firings. The projectile was launched at degree
measures in increments of 10, starting from zero and going to 80. The resultant data set is as
follows:
Angle(degrees) Range(cm)
0
148.8
10
150.3
20
166.1
30
189
40
178.1
50
149.7
60
137.1
70
96.3
80
48.8
This data was put into a scatter plot with a second order polynomial best fit curve. The plot is as
follows:
Angle Vs Range
Range
200
100
y = -0.0486x2 + 2.7856x + 139.1
R² = 0.9642
0
0
20
40
60
Angle
80
100
The equation of the line of best fit is y=-.0486X2+2.7856X+139.1
The angle used for our part II (45 degrees) was plugged into the best fit equation. The result was
X=166.04m
This range was compared to our experimental range of 1.625 m by using percent difference.
1.6604 − 1.625
(100)
1.6604
2% error
Using the graph to locate the corresponding angles for the ranges of 160 cm, 150 cm, and 140 cm
gave:
48.44 degrees and 8.88 degrees, 53.1 degrees and 4.22 degrees, 56.99 degrees and .32 degrees
respectively.
Conclusions
From this lab it became obvious that the angle of a launch has an affect on the range of
the projectile. The projectile will reach its maximum range at 45 degrees. The greater angles
cause a greater Y value. It was also learned that projectile experiments can be done without
measuring time. This was unknown and makes future experiments easier. The interplay of the
kinematic equations was elaborated on. Many values used, particularly time, were solved for in a
different equation before being reused in a new equation. The percent errors are a result of air
resistance, but more significantly by experimenter error. The projectile launcher was not fastened
to the table and therefore wiggled about when loading and firing. The cardboard on the floor may
have been kicked. The meter stick may not have been perfectly lined up under the opening to the
barrel. All of these contribute to the error.
Part 1
1.) Why are you to measure the bottom of the ball and not the middle?
The ball has no defined bottom, but as it is in the air it does not matter if it spins. This is
because when it strikes the ground it will strike on a position equidistant to the center as the
“bottom” of the ball was to the center before launch. This allows a very accurate result, as the
first thing that hits the ground is the “bottom” of the ball.