
AN INTRODUCTION TO NEWTONIAN
MECHANICS by Edward Kluk Dickinson State University, Dickinson ND 
A technical
introduction
This applet simulates all kinds of projectile
motion with low gravity acceleration and makes possible to time these motions.
A motion can start from eleven marked heights at the distance 0 m. Its initial
speed and direction are selectable from three labelled choice boxes. If default
sign "+" is selected, a selected projection angle counts counter clockwise
from a dotted horizontal line. Otherwise it counts clockwise. Every time
the red object reaches the edge of the vision field it stops and Reset button
is activated. This button let us restore initial conditions for the last
motion. The clear button restores default initial conditions for the motion
and repaints the vision field. Finally, if the path tracer is on, a path
of the object is drawn in the vision field.
How vertical and horizontal motions
add
We already know mathematical models for free
fall and undisturbed motion on a frictionless and impenetrable horizontal
plane that eliminates a gravitational pull . A natural question arises what
will happen if a body starts with nonzero horizontal speed without a support
of the horizontal impenetrable plane. A simple experiment with a body sliding
with some horizontal speed off a table shows the body motion as a combination
of vertical and horizontal motions. As we already know an accurate timing
of this kind of motion is not possible without high tech devices like a fast
photography. Some camcorders have fixed time between consecutive frames and
they can be used to study this kind of motion. But even with such camcorder
our experiment would be complicated.
It is much simpler to go back to our fictitious
planet with the very low gravity acceleration simulated in the applet. Start
it with all default settings except of speed which should be set at 0.20
m/s. This will simulate a case of the motion of our current interest.
Measure all time intervals the moving object needs to reach each consecutive
vertical dotted line from its starting point. These consecutive dotted lines
are in a distance of 1 m from each other. Therefore in the first measured
time interval the object covers 1 m in horizontal direction , in the
second interval 2 m, and so for. A total distance covered by the object for
each interval is greater than that because it moves simultaneously in both
horizontal and vertical directions. Graphing horizontal distance versus time
you will get exactly the same result as for the motion along a horizontal
frictionless plane. Namely, the horizontal component of object's speed
v_{H} remains constant because your graph is a straight
line. The slope of this line represents the mentioned above horizontal component
of the speed. Let us introduce a two dimensional Cartesian frame of reference
with an origin at zero height and zero displacement which is in the lower
left corner of the vision field, x axis directed horizontally
to the right and y axis directed vertically up. Then, you
should be able to conclude from your graph that the elapsed time
t and x coordinate of the object in the Cartesian
reference frame are related by the following formula
representing a uniform motion (motion with a constant speed and no
change of direction) along x axis.
Now it is time to collect next set of data related
to change of object's vertical position. Measure all time intervals the moving
object needs to reach each consecutive horizontal dotted line from its starting
point. These consecutive dotted lines are also in a distance of 1 m from
each other. Relying on your experience with a free fall motion you may expect
that a covered vertical distance d may be proportional to
t ^{2} rather than t. Graph
d versus t ^{2} looking for "experimental"
confirmation of this hypothesis. Your results including value of acceleration
should be practically identical like for the free fall case. After all you
are on the same low gravity acceleration planet. Therefore
where the acceleration a should be about 0.01 m / s
^{2}.
Math helps to reach more conclusions
The vertical distance d covered
by the falling object is related to its y coordinate. If an
initial y coordinate is denoted as
y_{o} then
Mathematical forms of x and y as functions
of time deduced from our "experimental" results show that the motion of the
object is composed of two independent simple motions. The horizontal motion
with a constant speed v_{H} and vertical free fall
with a constant acceleration a. They are independent in this
sense that a modification of the horizontal motion by changing its initial
speed v_{H} does not influence the vertical motion
and a modification of the vertical motion by changing its acceleration does
not influence the horizontal motion. To check "experimentally" the first
of these two properties of this kind of motion, make timing for a vertical
component of motions with initial horizontal speeds 0.1 m / s
and 0.15 m / s. If horizontal and vertical components of these
motions are independent then the vertical timings for both selected
motions should be the same within of an experimental accuracy. You should
be aware that if an air resistance plays an important role the horizontal
and vertical components of motion are not independent anymore.
Calculating t from the formula
for x and substituting the result to the formula for
y gives the following result
In the defined above Cartesian reference frame this relation formally describes
a parabola with opening down and vertex at x = 0 and
y = y_{o}. The upper part of the right branch of this parabola
describes the path of the object. To visualize it, run the applet with
the path tracer on and other parameters as defaults. Surprisingly enough,
Galileo knew that paths for this kinds of motion are parabolic.
Now insert into the parabola equation for y_{o} and
v_{H} their default values 10 m and
0.25 m / s . Select a few simple values for
x and calculate respective y values comparing them
with values for the same x values along the trace left by the
object. If our mathematical model of horizontal projectile motion is correct
the calculated and "experimental" y values should be in a good
agreement.
Evaluation
If at this point you do understand:
the objectives of this lesson are fully achieved. If you have doubts try to read it once more concentrating on them, but do not try to memorize this text. Physics is not about memorizing, it is about understanding.
Last update: Jan 10, 1997  E  mail to Edward Kluk 
Copyright (c) 1996 Edward Kluk 