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APSC 254
MEASUREMENT, MODELING AND UNCERTAINTY
O
BJECTIVES
Teach students the role of modeling in experimental science and the examination of
experimental uncertainties.
T
HEORY
Modeling of experimental results can be conducted in two ways.
The most
comprehensive method is when the model is developed from “first principles” i.e. from
the fundamental physical, chemical, or biological properties of a system.
With this type
of a model, the mathematical basis for the model is developed, predictions are made from
this, and these are tested experimentally.
The second method of model development is to
perform experiments and then develop a model that can be used to describe the data.
It is
incumbent on the experimenter to then refer the components of any equation that is used
as a “curve fit” back to fundamental system properties thus creating a model.
The
difference between a curve fit (a mathematical equation that can be used to describe a
trend in data) and a model is the understanding of the basis for the variables in the model.
In this exercise, we are examining projectile motion.
The fundamental principles
describing projectile motion are well understood.
The two basic parameters are the
velocity and the angle of the launch.
The horizontal range,
Δ
x
, for a projectile can be found using the following equation:
t
v
x
x
=
Δ
(1)
where
v
x
is the horizontal velocity and
t
is the time of flight.
To find the time of flight,
t
, the following kinematic equation is needed:
t
v
gt
y
y
0
2
2
1
+
=
Δ
(2)
where
Δ
y
is the height,
g
is the acceleration due to gravity and
v
y0
is the vertical
component of the initial velocity.
When a projectile is fired horizontally (from a height), the time of flight can be found
from rearranging Equation 2. Since the initial velocity is zero, the last term drops out of
the equation yielding:

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