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4.2 Determination of trajectories
4.2.1 Experiment setup
•
One possible experiment setup is shown schemati-
cally in Fig. 3 (not to scale). The drill holes in the
back plate of the pendulum are placed so that
when a projectile is fired to land directly on the
workbench, the launch heights are 50, 100, 150,
200 and 300 mm.
1
2
1
Fig. 3: Experiment setup, key:
Projectile launcher,
3
4
projectile,
Paper,
Carbon paper,
ample)
•
When launching against a vertical wall (e.g.
whiteboard U10030 mounted on easel U10381)
the radius of the projectile (1.25 cm) should be
subtracted from the distance between the point
of launch and the wall to obtain the distance
measurement x
. The height measurement y
M
tive to the launch height is given by the height of
the impact on the wall minus 62.5 mm, 112.5 mm,
162.5 mm, 212.5 mm or 312.5 depending on the
hole used.
4.2.2 Experiment procedure
•
It is practical for these experiments to note the
experiment number, the spring tension (1, 2 or
3), the launch angle and the values x
Example:
No
Spring
Launch-
tension
angle
ϕ / °
1
1
0
2
2
0
3
3
0
4
1
0
5
2
0
6
3
0
3B Scientific GmbH • Rudorffweg 8 • 21031 Hamburg • Germany • www.3bscientific.com • Technical amendments are possible
3 4
5
2
Launch position of
5
Easel with whiteboard (for ex-
rela-
M
and y
.
M
M
Projectile
Target
distance
height
x
/ cm
y
/ cm
M
M
171.3
–30
125.4
–30
86.9
–30
62.3
–15
90.5
–15
120.7
–15
4.1.3 Experiment evaluation
•
It is practical to take as the origin of the coordi-
nate system the mid-point of the projectile at the
moment of launch. Then the following applies:
cos ϕ
v
= v
X
0
sin ϕ
v
= v
Y
0
1
=
−
y v t
Y
2
x = v
t
X
From Eq. 20 t = x / v
eliminated from Eq.19.
•
If v
and v
are then eliminated from the resulting
X
y
equation using Eqs. 17 and 18, the following is
obtained
=
ϕ
−
2
y x
tan
x
This is the equation for the trajectory.
•
In this equation only the launch velocity v
known since the distances x and y were measured
during the course of the experiments. If v
culated for the various experiments, the follow-
ing results are obtained:
Spring tension
1
2
3
•
The numbers are based on a total of 25 experi-
ments, of which only 6 are explicitly listed in the
above table. The trajectory can now be obtained
from these using Eq. 21 and compared to the
measured values. The result is shown in Fig. 4.
Fig. 4: Comparison of measurements and calculated curve, x = horizontal
projectile distance, y = vertical height, symbols = measured values
(circles = spring tension 1, squares = spring tension 2, rhombuses = spring
tension 3), lines = calculated trajectories
8
(17)
(18)
2
gt
(19)
(20)
, whereby the time can be
X
g
(21).
ϕ
2
2
2
v
cos
0
v
in m/s
0
3.53
5.10
6.85
is un-
0
is cal-
0
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