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position (coordinate) functions x(t), y(t) and z(t) to define the position of an object in three-dimensional space, using their Cartesian coordinates as functions of time t;
velocity functions x'(t), y'(t) and z'(t) to define the rate of change of position of an object in three-dimensional space; these functions are first derivatives of corresponding position functions;
acceleration functions x"(t), y"(t) and z"(t) to define the rate of change of velocity of an object in three-dimensional space; these functions are second derivatives of corresponding position functions.
In general, to graphically analyze any function y=f(x) of a single argument x, we use two-dimensional plane with XY-coordinates and construct a graph of this function as a set of all points {x, y=f(x)} for all arguments x in the domain of this function.
We will do this analysis for each characteristic of motion as function of time argument t (position, velocity and acceleration) in some simple cases.
Our main case is a motion along the straight line. It simplifies our
analysis, as we can choose an X-axis coinciding with the line of motion,
and Y- and Z-coordinates of an object will always be zero. So, we will
have only one set of functions - x(t), x'(t) and x"(t) to analyze.
We will also extend the time line to both directions, positive (future)
and negative (past), because our null-moment is chosen arbitrarily.
Our first case is an object at rest. Its X-coordinate is constant: x(t)=5. The first and second derivatives of this function are zero: x'(t)=0, x"(t)=0. Graphically, all three characteristics of motion look like this:
Next let's consider a uniform motion along an X-axis, according to position function x(t)=5+3t. Its velocity function (the first derivative of position) is x'(t)=3 (constant for uniform motion) and its acceleration (second derivative) is x"(t)=0. Graphically, all three characteristics of motion look like this:
Now let's consider a case of an object falling from some height down along an X-axis, according to position function x(t)=3−t²/2. Its velocity function (the first derivative of position) is x'(t)=−t and its acceleration (second derivative) is x"(t)=−1 (constant). Graphically, all three characteristics of motion look like this:
Finally, let's consider a case of an object oscillating left and right
along an X-axis around point 0, according to position function x(t)=sin(t). Its velocity function (the first derivative of position) is x'(t)=cos(t) and its acceleration (second derivative) is x"(t)=−sin(t). Graphically, all three characteristics of motion look like this:
Recall that, if x(t), y(t) and z(t) are coordinate functions of the moving object, the derivatives of these functions x'(t), y'(t) and z'(t) represent components of a velocity of this moving object, that is a vector of instantaneous change of an object's position.
Let's generalize. If function h(t) represents the value of some physical characteristic as a function of time, its derivative h'(t) represents an instantaneous speed of change of this physical characteristic.
Using this logic, we can find an instantaneous speed of change of an
instantaneous speed of change, which would be a derivative of the
derivative, that is a second derivative of a base function h(t).
This is the logic behind analyzing how an instantaneous speed of change of an object's position itself changes with time.
Acceleration is an instantaneous speed of change of an
instantaneous speed of change of an object's position. By definition, it
is a vector with components equal to derivatives of corresponding
components of an instantaneous speed of change of an object's position.
Considering the latter is a vector with components equal to derivatives
of original coordinate functions, acceleration is a vector a represented by three second derivatives of the coordinate functions describing the object's position:
a = {x"(t), y"(t), z"(t)}
A short note about terminology.
We distinguished velocity (a vector) from speed (its magnitude). In case of acceleration there is no separate word to distinguish a vector of acceleration from its magnitude. Both are called acceleration, and it's the context that helps to distinguish one from another.
Usually, if the motion occurs in one dimension along a straight line in
one direction, there is no big difference between a vector and its
magnitude, and there should be no confusion. In two- and
three-dimensional cases with movement along some curve we will usually
consider acceleration as a vector.
Let's consider a few examples.
Example 1. An object is at rest at the origin of coordinates.
So, x(t)=y(t)=z(t)=0 for all time moments t.
Velocity v of this movement is null-vector for all time moments, since a derivative from a constant equals to 0.
Indeed, x'(t)=y'(t)=z'(t)=0 and v = {x'(t), y'(t), z'(t)} =
= {0, 0, 0}.
Hence, its instantaneous speed s (magnitude of the vector of velocity) is also 0.
Acceleration, similarly, is a null-vector for all time moments as well, as a = {x"(t), y"(t), z"(t)} =
= {0, 0, 0}.
Its magnitude is, obviously, 0.
Example 2. An object is at rest at a point with coordinates {3, 10, −6}.
So, x(t)=3; y(t)=10; z(t)=−6 for all time moments t.
Velocity v of this movement is null-vector for all time moments, since a derivative from a constant equals to 0.
Indeed, x'(t)=y'(t)=z'(t)=0 and v = {x'(t), y'(t), z'(t)} =
= {0, 0, 0}.
Hence, its instantaneous speed s (magnitude of the vector of velocity) is also 0.
Acceleration, similarly, is a null-vector for all time moments as well, as a = {x"(t), y"(t), z"(t)} =
= {0, 0, 0}.
Its magnitude is, obviously, 0.
Example 3. An object is in the uniform motion. At time t=0 it is at a point with coordinates {3, 10, −6}.
It moves along a straight line according to the following coordinate functions:
x(t)=3+6·t;
y(t)=10−8·t;
z(t)=−6+10·t
for all time moments t.
Velocity v is vector with the following components:
x'(t)=6; y'(t)=−8; z'(t)=10.
Therefore, v = {x'(t), y'(t), z'(t)} =
= {6, −8, 10}.
Hence, its instantaneous speed s (magnitude of the vector of velocity) is
s = √(6)²+(−8)²+(10)² = 10√2.
Acceleration is a null-vector for all time moments of time because the velocity is constant:
a = {x"(t), y"(t), z"(t)} =
= {0, 0, 0}.
The magnitude of acceleration is, obviously, 0.
Example 4. An object falls down from the Tower of Pisa, pulled by gravity, accelerating all the time. At time t=0 it is at the top of the tower at a point with coordinates {0, 0, H}, where H - some positive number that defines initial height of an object - the height of the Tower of Pisa.
It moves along a straight line down the Z-axis towards point {0, 0, 0} on the ground, according to the following coordinate functions:
x(t)=0;
y(t)=0;
z(t)=H−g·t²/2
(where g is a positive constant related to strength of the gravity)
for all time moments t.
Velocity v is vector with the following components:
x'(t)=0; y'(t)=0; z'(t)=−g·t.
Therefore, v = {x'(t), y'(t), z'(t)} =
= {0, 0, −g·t}.
Hence, its instantaneous speed s (magnitude of the vector of velocity) is
s = √(0)²+(0)²+(−g·t)² = g·t.
Acceleration a is a vector with the following components:
a = {x"(t), y"(t), z"(t)} =
= {0, 0, −g}.
The magnitude of acceleration is, obviously, g.
We can also calculate the time, when this object reaches the ground (that is, when z(t)=0). For this we just have to resolve the equation z(t) = H−g·t²/2 = 0.
This is the first Physics lecture, where a solid mathematical background will be needed.
In particular, we assume the familiarity with vectors and derivatives.
These and many other mathematical concepts can be found in the course
"Math 4 Teens" on the same site as this lecture - UNIZOR.COM.
In the first approximation speed is a distance covered by a moving object in a unit of time.
What's wrong with this definition?
Firstly, a moving object moves with a different speed at different
moments in time, so speed is a function of time, which is not reflected
in this definition.
Secondly, what is a "unit of time"? Any unit of time, like a second, is
not a "moment" in time, but an interval, during which an object can
still change its speed.
Let's approach this concept more rigorously.
Our task is to define a speed as a characteristic of motion at any moment in time.
The simplest motion we consider first is a movement of an object along
the X-axis, according to the following function of X-coordinate of time t: x(t) = a·t - a uniform movement along the straight line (X-axis).
Let's define the average speed during any time interval from moment t1 to moment t2 as the distance covered during this time interval divided by the length of time this distance was covered.
For an object moving along the X-axis this average speed can be expressed in a formula:
save(t1, t2) =
= [x(t2) − x(t1)]/ (t2 − t1)
Remarkably, the average speed for an object uniformly moving along the X-axis as x(t) = a·t will be constant, independently of a choice of t1 and t2 - the beginning and ending moments in time:
save(t1, t2) =
= [a·t2 − a·t1]/ (t2 − t1) = a
Of course, this type of movement is called uniform exactly for this reason - average speed of movement is constant on any time interval.
Consider now a movement along the X-axis according to general function x(t) and the task of defining its speed at any moment t.
Let's note the position of our object at two moments in time - t and t+Δt. We can find an average speed during this time period:
save(t, t+Δt) =
= [x(t+Δt) − x(t)]/ (t+Δt − t) =
= [x(t+Δt) − x(t)]/Δt
It is reasonable now to define a speed at moment t, called instantaneous speeds(t), as a limit of the above expression for the average speed on interval [t, t+Δt], when Δt is diminishing to 0, that is time increment Δt is an infinitesimal variable:
s(t) =
= limΔt→0[x(t+Δt) − x(t)]/Δt
The limit above is a definition of the derivative of a function x(t). Therefore, the rigorous definition of an instantaneous speed of an object moving along the X-axis according to function of time x(t) is the derivative of this function:
s(t) = dx(t)/dt = x'(t)
In one of the previous lectures we mentioned that coordinate functions x(t), y(t) and z(t) must be continuous to prevent instant science fiction jumps to other planets. From now on we will also assume that these coordinate functions are differentiable to be able to determine the speed of movement as a derivative of a coordinate function by time.
This completes the definition of speed for a one-dimensional movement along the X-axis.
Now we will address general case of movement in three-dimensional space described by coordinate functions x(t), y(t) and z(t).
To define a speed in case of general three-dimensional movement represented by three coordinate functions x(t), y(t) and z(t), we will use the vector representation of the position of an object.
So, assume that at time from t to t+Δt a vector a from the origin of coordinates to point {x(t), y(t), z(t)} is transformed into vector b from the origin of coordinates to point {x(t+Δt), y(t+Δt), z(t+Δt)}.
Then the difference between these vectors is a displacement that
happened with a moving object during this time. The first important
thing to notice here is that this displacement is a vectord = b − a.
Though we did not mention the vector character of the movement along the
X-axis above, it was implicitly there, because movements forward
(towards increasing X-coordinates) or backward (towards decreasing
X-coordinates) are two different directions that are characteristic of
vectors. So, the positive or negative displacement [x(t+Δt) − x(t)] is really a vector because it has a sign - positive if x(t+Δt) > x(t) and negative if x(t+Δt) < x(t), and a sign in one-dimentional case of movement along the X-axis is the direction.
Now, when we have determined the vector character of a displacement from a position a at time t to position b at time t+Δt, we can define the average displacement during this period of time - it's a vector(b − a) /Δt.
Coordinates of this vector are, obviously, {Δx(t)/Δt, Δy(t)/Δt, Δz(t)/Δt}, where
Δx(t) = x(t+Δt) − x(t),
Δy(t) = y(t+Δt) − y(t),
Δz(t) = z(t+Δt) − z(t).
When time increment Δt is infinitesimal, all three components of the displacement vector are infinitesimal as well.
If all three coordinate functions are differentiable (which we will always assume), the limit of the average displacement vector(b − a) /Δt on a time interval Δt, when Δt→0, is a vector with coordinates {x'(t), y'(t), z'(t)}.
Hence, the instantaneous speed of movement of an object defined by coordinate functions x(t), y(t) and z(t) is a vector with coordinates {x'(t), y'(t), z'(t)}.
This vector is called velocity, while the term speed is reserved only for the magnitude of this vector, that is its absolute value, disregarding the direction, which is equal to
This is the first Physics lecture, where a solid mathematical background will be needed.
In particular, we assume the familiarity with vectors and derivatives.
These and many other mathematical concepts can be found in the course
"Math 4 Teens" on the same site as this lecture - UNIZOR.COM.
In the first approximation speed is a distance covered by a moving object in a unit of time.
What's wrong with this definition?
Firstly, a moving object moves with a different speed at different
moments in time, so speed is a function of time, which is not reflected
in this definition.
Secondly, what is a "unit of time"? Any unit of time, like a second, is
not a "moment" in time, but an interval, during which an object can
still change its speed.
Let's approach this concept more rigorously.
Our task is to define a speed as a characteristic of motion at any moment in time.
The simplest motion we consider first is a movement of an object along
the X-axis, according to the following function of X-coordinate of time t: x(t) = a·t - a uniform movement along the straight line (X-axis).
Let's define the average speed during any time interval from moment t1 to moment t2 as the distance covered during this time interval divided by the length of time this distance was covered.
For an object moving along the X-axis this average speed can be expressed in a formula:
save(t1, t2) =
= [x(t2) − x(t1)]/ (t2 − t1)
Remarkably, the average speed for an object uniformly moving along the X-axis as x(t) = a·t will be constant, independently of a choice of t1 and t2 - the beginning and ending moments in time:
save(t1, t2) =
= [a·t2 − a·t1]/ (t2 − t1) = a
Of course, this type of movement is called uniform exactly for this reason - average speed of movement is constant on any time interval.
Consider now a movement along the X-axis according to general function x(t) and the task of defining its speed at any moment t.
Let's note the position of our object at two moments in time - t and t+Δt. We can find an average speed during this time period:
save(t, t+Δt) =
= [x(t+Δt) − x(t)]/ (t+Δt − t) =
= [x(t+Δt) − x(t)]/Δt
It is reasonable now to define a speed at moment t, called instantaneous speeds(t), as a limit of the above expression for the average speed on interval [t, t+Δt], when Δt is diminishing to 0, that is time increment Δt is an infinitesimal variable:
s(t) =
= limΔt→0[x(t+Δt) − x(t)]/Δt
The limit above is a definition of the derivative of a function x(t). Therefore, the rigorous definition of an instantaneous speed of an object moving along the X-axis according to function of time x(t) is the derivative of this function:
s(t) = dx(t)/dt = x'(t)
In one of the previous lectures we mentioned that coordinate functions x(t), y(t) and z(t) must be continuous to prevent instant science fiction jumps to other planets. From now on we will also assume that these coordinate functions are differentiable to be able to determine the speed of movement as a derivative of a coordinate function by time.
This completes the definition of speed for a one-dimensional movement along the X-axis.
Now we will address general case of movement in three-dimensional space described by coordinate functions x(t), y(t) and z(t).
To define a speed in case of general three-dimensional movement represented by three coordinate functions x(t), y(t) and z(t), we will use the vector representation of the position of an object.
So, assume that at time from t to t+Δt a vector a from the origin of coordinates to point {x(t), y(t), z(t)} is transformed into vector b from the origin of coordinates to point {x(t+Δt), y(t+Δt), z(t+Δt)}.
Then the difference between these vectors is a displacement that
happened with a moving object during this time. The first important
thing to notice here is that this displacement is a vectord = b − a.
Though we did not mention the vector character of the movement along the
X-axis above, it was implicitly there, because movements forward
(towards increasing X-coordinates) or backward (towards decreasing
X-coordinates) are two different directions that are characteristic of
vectors. So, the positive or negative displacement [x(t+Δt) − x(t)] is really a vector because it has a sign - positive if x(t+Δt) > x(t) and negative if x(t+Δt) < x(t), and a sign in one-dimentional case of movement along the X-axis is the direction.
Now, when we have determined the vector character of a displacement from a position a at time t to position b at time t+Δt, we can define the average displacement during this period of time - it's a vector(b − a) /Δt.
Coordinates of this vector are, obviously, {Δx(t)/Δt, Δy(t)/Δt, Δz(t)/Δt}, where
Δx(t) = x(t+Δt) − x(t),
Δy(t) = y(t+Δt) − y(t),
Δz(t) = z(t+Δt) − z(t).
When time increment Δt is infinitesimal, all three components of the displacement vector are infinitesimal as well.
If all three coordinate functions are differentiable (which we will always assume), the limit of the average displacement vector(b − a) /Δt on a time interval Δt, when Δt→0, is a vector with coordinates {x'(t), y'(t), z'(t)}.
Hence, the instantaneous speed of movement of an object defined by coordinate functions x(t), y(t) and z(t) is a vector with coordinates {x'(t), y'(t), z'(t)}.
This vector is called velocity, while the term speed is reserved only for the magnitude of this vector, that is its absolute value, disregarding the direction.
Time is an undefined concept. In this way it is similar to such concepts as geometric point. It is specifically physical concept, and Physics needs this concept to quantitatively characterize the physical processes.
As is the case with any mathematical undefined concepts, we study not the concept of time itself, but its properties that we postulate.
Though time-related postulates might not seem as mathematically
rigorous as Euclidean postulates in Geometry, we will try to describe
them with utmost accuracy.
First of all, time is one of the forms of existence of the world around us, and any change in the world are related to change of time.
There are no changes in the world not related to change of time and,
from the opposite side, there is no change in time without some changes
in the world.
Using this connection between change of time and some changes in the
world around us, we can choose one particular process that occurs in our
world with relative regularity as the main time-measuring device and
measure time by changes in this process. Obviously, this process should
be stable, repetitive, regular, predictable etc. to be used as a
time-measuring device.
So, what can be used as such a process?
In the previous lecture we have suggested that the rotation of the Earth
around its axis can be used as this process. We have divided the period
of one rotation into 24 hours, each hour - into 60 minutes, each minute
- into 60 seconds, and suggested a second as the main unit of time.
In addition, classical Physics (a subject of this course) assumes (postulates, makes it an axiom) the continuity of time,
which implies that we can divide any interval of time into however
small intervals and still obtain valid time intervals, reflecting
certain changes in the world. That's why we can talk about milliseconds
(1/1000th of a second), microseconds (1/1000000th of a second) etc.
That means that we can choose any arbitrary moment of time as the beginning of time
(zero point) and use a real number of seconds since or before this
moment to any other moment of time. So, time can be measured and any
moment of time can be characterized by a real number - the number of
second since or before the beginning of time to this moment. This real
number is positive for all moments of time that characterize the
processes happened after the beginning of time and it is negative for
those that precede the beginning.
But how to determine the period of one rotation of the Earth that we
suggested as a time-measuring device? Well, we can use a telescope fixed
at some place on Earth, look at the stars and watch how they change
their location in the telescope. Obviously, they will move, as our
planet rotates, and after one rotation the stars will be in the same
position on the sky and in the telescope.
Unfortunately, this might be a good measure for technology in Ancient
Egypt, but today's necessities need more precision. So, let us describe
the contemporary time-measuring process.
Atomic clock is considered nowadays a standard for precise time
measurement. The time interval of 1 second in the International System
of Units (SI) is derived from the oscillation between two states of an
atom of an element cesium. More exactly, 1 second is a time interval
during which 9,192,631,770 oscillations occur. This clock's precision is
1 second in about 30 million years - quite sufficient for all
foreseeable needs.
For all simpler practical reasons people use the clock as a time
measuring device, periodically synchronizing it with atomic clock
through different interfaces.
We have already discussed the first axiom of time - continuity.
There is another very important time-related axiom accepted by classical
Physics. It states that physical processes behave the same, regardless
of time when they occur. In other words, an experiment conducted today
will have exactly the same outcome as an identical experiment conducted
tomorrow. In this statement the word "identical" is very important, it
means that everything involved in the experiment today must be the same
as in the tomorrow's experiment. If this rule is observed, the only
difference between experiments is the time when they are conducted, and
that must not affect the results of an experiment.
Another form of this axiom is: time is uniform.
CONCLUSION
The continuity and uniformity are properties of an abstract concept of time
that we use as the characteristic of all processes occurred in our
world. Time intervals can be measured by different kinds of clocks, the
unit of measurement accepted in the International System of Units (SI)
is a second.
Using these properties of time to define the motion, we can
always describe a motion in our three-dimensional space as three real
functions (space coordinates) x(t), y(t) and z(t) of real argument (time) t. The only thing we need is a system of Cartesian coordinates and a moment of time we choose as the beginning of motion.
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.
When talking about motion, we have to start by defining two things:
what is moving (moving object) and
what does it mean that this object is moving.
These are qualitative characteristics.
When these issues are clarified, the next issue is to describe the
characteristics of the moving object and to describe in certain terms
the parameters of the movement.
These are quantitative characteristics.
Then, knowing quantitative characteristics of the moving object and the
parameters of its movement, our task is to come up with certain Laws of
Motion. These laws are supposed to be universal and independent of the
characteristics of the moving object and parameters of its movement.
Let's examine the moving object. In this course in most cases we will assume that the moving object is a geometric point in three-dimensional space. Therefore, the size of a moving object is zero in all directions.
Before defining the motion, we have to introduce another physical concept - time.
First of all, time is an undefinable characteristic of all processes taking place in our world. The fact that time
is undefinable is not a good news, but the good news is that we can
measure it by comparing any process with some standard process and
calculating the rate of any process in units of comparison of it with
the rate of a standard process.
Since our moving object is a point in three-dimensional space, we can
consider some system of coordinates in this space and talk about a position
of our moving object (point) in terms of its coordinates. So, three
coordinates of our moving object in this system of coordinates determine
its position.
Now we can define a concept of motion as a process (the rate of which can be measured by time) during which the position of our moving object (a point with three coordinates) is changing with time.
If M is a moving object, we can talk about its position at any moment of time t defined by three coordinates - functions of time x(t), y(t) and z(t).
We have come up with a conclusion that motion can be represented as three functions of time - the X-, Y- and Z-coordinates of a point that represents our moving object.
Since any function, in mathematical terms, represents a transformation of one number (in our case, time) into another (in our case, coordinate), we have to understand what kind of numbers (time arguments and coordinate values) we are dealing with.
Let's start with time.
To talk about numbers that represent time, we have to know how to
measure time. We need to measure the time interval, so we need to know
when the time starts, which in many cases is a moment the motion starts,
and how long it lasts from one moment (say, from the beginning of
motion) to another moment (for example, a moment of observation). That
requires the unit of time to measure this interval.
Let's assume that we have chosen an interval of time of one rotation of our planet Earth around its axis to be equal to 24 hours, each hour to be equal to 60 minutes and each minute to be equal to 60 seconds. So, the interval of time of one rotation of Earth equals to 24 hours or 24·60=1440 minutes or 1440·60=86400 seconds.
From now on in most cases we will use seconds as the main unit of
time because this is an international standard. Having the unit of time
defined, we can talk about the numerical time argument in coordinate
functions x(t), y(t) and z(t).
Now let's consider what values these coordinate functions take.
This is a familiar Cartesian coordinates, so all we need is a system of
coordinates. But this is exactly the problem. Which system of
coordinates should we choose and which unit of length to use?
The answer to this question is not easy. Under typical conditions we can
say that the system of coordinates has its origin at the point of the
beginning of the motion, which means that coordinates of our moving
object (point) at time t=0 are equal to 0, that is x(0)=0, y(0)=0 and z(0)=0.
As for direction of the axis of coordinates, we might choose any three
orthogonal directions, but usually will choose them to simplify the
coordinate functions.
For example, if we are dealing with a straight line movement, we better
choose X-axis along the direction of this movement, which leaves only
one coordinate function, x(t), to represent the movement, since other coordinate functions, y(t) and z(t) will always be equal to zero for any time moment t.
Finally, the unit of length should be chosen, and there are many. In most cases we will use metric meter as a unit of length along each coordinate axis, because it's an international standard.
With a choice of the beginning of time, the unit of time measurement,
the coordinate system and the unit of length we have fully defined all
the components needed to meaningfully describe the motion as represented by three coordinate functions x(t), y(t) and z(t), where time t is an argument and function values define the position of a moving object (point) in our three-dimensional space at time t.
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