*Notes to a video lecture on http://www.unizor.com*

__Graphical Representation__

of Motion

of Motion

We have studied three categories of functions:

*position*(coordinate) functions

**,**

*x(t)***and**

*y(t)***to define the position of an object in three-dimensional space, using their Cartesian coordinates as functions of time**

*z(t)***;**

*t**velocity*functions

**,**

*x'(t)***and**

*y'(t)***to define the rate of change of**

*z'(t)**position*of an object in three-dimensional space; these functions are first derivatives of corresponding

*position*functions;

*acceleration*functions

**,**

*x"(t)***and**

*y"(t)***to define the rate of change of**

*z"(t)**velocity*of an object in three-dimensional space; these functions are second derivatives of corresponding

*position*functions.

In general, to graphically analyze any function

*of a single argument*

**y=f(x)****, we use two-dimensional plane with XY-coordinates and construct a graph of this function as a set of all points**

*x***}**

*x, y=f(x)***in the domain of this function.**

*x*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)***and**

*x'(t)***to analyze.**

*x"(t)*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:

**. The first and second derivatives of this function are zero:**

*x(t)=5***,**

*x'(t)=0***. Graphically, all three characteristics of motion look like this:**

*x"(t)=0*Next let's consider a uniform motion along an X-axis, according to position function

**. Its velocity function (the first derivative of position) is**

*x(t)=5+3t***(constant for uniform motion) and its acceleration (second derivative) is**

*x'(t)=3***. Graphically, all three characteristics of motion look like this:**

*x"(t)=0*Now let's consider a case of an object falling from some height down along an X-axis, according to position function

**. Its velocity function (the first derivative of position) is**

*x(t)=3−t²/2***and its acceleration (second derivative) is**

*x'(t)=−t***(constant). Graphically, all three characteristics of motion look like this:**

*x"(t)=−1*Finally, let's consider a case of an object oscillating left and right

along an X-axis around point 0, according to position function

**. Its velocity function (the first derivative of position) is**

*x(t)=sin(t)***and its acceleration (second derivative) is**

*x'(t)=cos(t)***. Graphically, all three characteristics of motion look like this:**

*x"(t)=−sin(t)*
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