Experiment 3: Projectile

 
Objective
To study the ballistic motion of a ball under the influence of gravity as a function of
the launch angle.
Theoretical Background
An object launched into the air at the surface of the earth is known in physics as a
projectile. Its motion is called a ballistic motion. Once the projectile is launched, it
moves under the influence of the earth’s gravitational force and of forces exerted
by the air. Here we will neglect the effects due to the air. During the launch process
itself, other forces are exerted on the object, usually for a short period of time,
which result in a certain initial velocity, initial height and a certain launch angle.
With the knowledge of those initial conditions, it is possible to calculate the
complete trajectory of the projectile. Among other quantities, one can compute its
range (horizontal distance travelled), its maximum height, and its time of flight. In
this experiment, we will focus on the range.
The general trajectory of the projectile can be understood thanks to the examples
studied in the first two experiment (“Understanding motion” and “Free fall”). It is
instructive to decompose the motion along the horizontal (x-axis) and the vertical
(y-axis). As gravity acts only along the vertical, the vertical motion is a constant
acceleration motion with acceleration g pointing downwards. This is the situation
studied in the “Free Fall” experiment. The horizontal motion on the other hand is a
constant velocity motion, since there are no forces, thus no acceleration along the
x-axis.
We choose a coordinate system with the positive y-axis pointing upwards:
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The constant acceleration is then a0  g , and the y-components of the motion
are given by:
2
0
1 2
y(t)  y  voyt  gt Eq. (1)
vy(t)  voy  gt

ay (t)  g y0  y(t  0) and v0 y  vy (t  0) are the vertical components of the position and of
the velocity of the projectile at time zero. Here, the time at which the projectile is
released is taken as time zero.
The horizontal component of the motion is given by:
x t x v t
( )  0  ox

vx(t)  vox ax(t)  0 x0  x(t  0) and v0x  vx(t  0) are the horizontal components of the position and
of the velocity of the projectile at time zero.
The launch angle  (measured from the horizontal) and the initial velocity v0
determine the values of
v0x and v0 y :
v0x  v0 cos()

v0 y  v0 sin()
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In this experiment, we will measure the initial velocity of a projectile ball v0 , the
launch angle  and the horizontal distance travelled between by the ball before it
hits the ground, known as the range R  x(t f )  xo , where t f is the time at which
the ball hits the ground and therefore terminates its projectile motion.
Considering the case where the initial and final heights are the same, the equations
lead to:
g
v
R
0 2 sin(2)
 Eq. (9)
Equipment
Power supply, Ballistic launcher unit with velocity meter, Meter rod, Carbon paper
Experimental Background
In this experiment, we use a wooden ball as projectile. It is launched by the ballistic
unit, which consist of a spring launcher with adjustable angle and a velocity meter.
The system allows loading the spring with different tensions. We will use the first
position (giving lowest initial velocity). When the spring is released, the velocity
meter records the initial velocity of the ball and the angle can be read from the
markings on the ballistic unit. This determine  and v0 . The ball falls on a bench
covered with carbon paper and produces a mark on it. The height at which the ball
falls is the same as the initial launching height. The range of the projectile can then
be measured with a ruler and compared to Eq. (7). The procedure is repeated two
more times for each launching angle and a series of angles are used as given in
Table 3.1.
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Procedure
1. Listen to and follow the instructions of the instructor very carefully. Do
not place your fingers in the launching area !
2. The experimental setup is as shown in Figure 3. 1
Figure 3. 1: Projectile launcher experimental setup (1)
3. Make sure the ballistic unit is aligned with the bench and that the bench is
covered with carbon paper. There should be no space between the unit and
the bench.
4. Adjust the angle according to the value given in Table 3.1.
5. Record the angle and its uncertainty.
6. Load the spring in the first position.
7. Release the spring, the ball will start its projectile motion.
8. Record the initial velocity, as given by the velocity meter.
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9. Find the mark made by the ball on the carbon paper.
10.Measure the horizontal distance travelled by the ball (range).
11. Repeat the steps 6-10 two more time.
12.Change the launch angle according to Table 3.1.and repeat the steps 4-11.
Table 3. 1: Data table
Data Analysis
1. Compute the average initial velocity v0av from the 3 trials at each angle.
2. Calculate its uncertainty using the standard deviation of the mean method.
3. Calculate the average initial velocity of all trials and its uncertainty.
4. Compute the average range Rav from the 3 trials at each angle.
5. Calculate its uncertainty using the standard deviation of the mean method.
6. Calculate the value of sin(2) for each angle, then its uncertainty using the error
propagation method.
7. Plot a graph sin(2) (y-axis) Vs. Rav (x-axis).
8. Add error bars to your graph.
9. Fit your data; get the slope, the y-intercept, and the fit quality parameter R2 .



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10.Find the uncertainties in the slope and intercept using the LINEST function in
Excel.
11.From the value of the slope and using Eq. (9) calculate the corresponding value
of
v0 and its uncertainty (use g =9.8 m/s2).
Table 2. 2: Data Analysis table
Discussion
1. Discuss whether you data shows a change in the initial velocity, when the
launch angle is changed. Give an explanation for what you observe
2. Discuss whether your graph is linear or not and why, according to the theory, it
should (or should not) be.
3. Compare the value of v0 obtained from the slope of the graph (step 11 above)
to the value calculated independently in step 3. (Do a quantitative comparison
and include uncertainties


sin(2) u(sin(2 ))
v0av u(v0 av )
Rav
u(Rav)

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References
1. PHYWE series of publications • Laboratory Experiments • Physics • P2131100
© PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Note
1) Make sure you understand the graph you have plotted and check which
distance corresponds to which initial angle in this graph.
2) The value for the range R can be derived as follows:
The condition is that
y(t)  y0 . Eq (1) therefore gives:
g
v
v g
t
v gt
v t gt
oy
oy
oy
2 2 sin( )
1 2
0
1 2
0
2

 

 
Where we used Eq.(8): voy  v0 sin() in the last equality. Plugging this value of