Wednesday, April 11, 2018

Unizor - Physics4Teens - Mechanics - Kinematics - Trajectory

Notes to a video lecture on


Trajectory is a set of all points in the three-dimensional space visited by a moving object (a point) during its motion. In other words, it's a trace of the motion.

If an object is moving, we can see its position at any moment in time.
After it has left one position in space and moved to another, the old
position is empty, we cannot see anything there, unless we leave certain
mark in space at every position visited by our object.

So, we cannot see an entire trajectory, unless we use some special
marking in our three dimensional space for each position visited by a
moving object.

For example, if our object is a tip of a pencil and its movement is
restricted to a surface of a paper, such a movement leaves a line on a
paper that represents the trajectory.

In three dimensional space it's more difficult to arrange conditions of a
motion that leaves a visible trace, but not impossible. Wilson Cloud
Chamber is an example of such an arrangement. Elementary particles
flying through it leave traces.

Now, when a concept of trajectory is introduced, let's define it more rigorously.

When talking about motion in more precise terms, we imply existence of
Cartesian coordinates with known origin, direction of axes and unit of
measurement of the length, certain time interval (usually, from 0 to some time limit T), during which the motion takes place, and coordinate functions of time x(t), y(t) and z(t), defining X-, Y- and Z-coordinates of a moving object at any moment of time t. These functions are defined on time interval [0, T], so we know the position of a moving object at any moment of time.

With that information we can define a trajectory of the movement as a set of all points with coordinates {x(t), y(t), z(t)}, where time parameter t belongs to an interval [0, T].

Obviously, the three coordinate functions x(t), y(t) and z(t) must be defined for any argument t that belongs to interval [0, T] (mathematically expressed as t ∈ [0, T]).

We can say even more about these functions. They must be continuous,
otherwise it would appear that in zero time our object moved from one
point to another on some distance, implying its infinite speed - a dream
of science fiction.

While we have no knowledge about forces affecting the motion, we cannot
predict the trajectory. All we can do is to observe it. Knowing all the
forces and the laws describing the results of their action on a moving
object, we will be able to predict the trajectory beforehand. This will
be discussed in lectures dedicated to Dynamics later in this course.

Let's discuss certain types of trajectories we will be dealing with in this course.

Straight line trajectory is the simplest one. Describing a motion
along this line, we will usually have the X-axis directed along the
line of motion. That results in Y- and Z-coordinates of a moving object
to be equal to zero at all times, that is y(t)=0 and z(t)=0 for all t ∈ [0, T].

So, only one function x(t), defined for all t ∈ [0, T] is sufficient to describe the motion along the straight line. For a motion in one direction (only forward) the function x(t) is monotonic. The "standing still" at the beginning of motion is described by function x(t) equal to zero for all values of time argument t. For oscillating movement around the origin along the X-axis function x(t) might resemble some kind of wave.

While qualitative analysis of function x(t) that describes certain types of straight line
movements is interesting, the reverse analysis is more fruitful from
the theoretical viewpoint. Let's examine a trajectory for a few types of
coordinate function x(t) (assuming x(0)=0 to signify that an object is at the origin of coordinates at time t=0 and two other coordinate functions, y(t) and z(t), are equal to zero for all t ∈ [0, T], which assures the movement along the straight line - the X-axis).

If x(t)=0 for all t ∈ [0, T], the object does not move from the origin of coordinates.

If x(t)=a·t, the object moves along the X-axis with constant speed, that is covering a units of length (say, meters) for each unit of time (say, second).

If x(t)=a·sin(t), the object moves along the X-axis oscillating back and forth.

If x(t)=2t−1, the object moves along the X-axis
always in a positive direction with increasing speed, that is covering
greater distance in a unit of time for greater values of time t. In other words, from t=100 to t=101 the object covers greater distance than from t=10 to t=11.

Circular trajectory occurs when an object moves within certain plane in our three-dimensional space in a circle of some radius R
around some central point. An example of such a movement might be the
rotation of any point on a carousel around its center. In this case it's
convenient to choose a system of Cartesian coordinates with origin in a
center of rotation with a plane of rotation lying in the XY-plane. Then
the Z-coordinate of a moving object will always be zero and the motion
can be described by two functions x(t) and y(t), always satisfying the equation x2(t) + y2(t) = R2.

For example, these coordinate functions define a motion with circular trajectory of radius R within an XY-plane with a center at the origin of coordinates:

x(t) = R·cos(t)

y(t) = R·sin(t)

z(t) = 0

Spiral trajectory occurs when an object moves in our three-dimensional space along a cylindrical surface of some radius R,
circling around the cylinder's axis with its projection on this axis
moving always forward. An example of this motion might be the movement
of the tip of a cork screw, when you open a bottle standing on a table.
While getting down into the cork, this tip makes small circles around
the cork screw's main axis. In this case it's convenient to choose a
system of Cartesian coordinates with Z-axis going down along the axis of
rotation of a cork screw (down into the bottle along its central axis)
and XY-plane to be perpendicular to this axis (and perpendicular to a
bottle's central axis), aligned to the top of a bottle. Then the
Z-coordinate of a tip of a cork screw will be a function of time
representing the depth of the tip inside a cork. The X and Y coordinates
will be similar to those for a plain circular trajectory described above.

For example, the following coordinate functions define a motion with spiral trajectory:

x(t) = R·cos(t)

y(t) = R·sin(t)

z(t) = a·t

Obviously, the complexity of the trajectories has no limits, but in this course we will be dealing with simple ones.

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