The Cartesian Coordinate System *Addendum*
At this point, it is better to talk about the directions of the velocity and acceleration vectors before moving on to curvilinear motion.
___Velocity____________
Here is a good graphic that describes the direction of the velocity vector.
Consider the vector \vec{r_i}. Denote i=t, the initial time and \vec{r_f} the vector at some later time f=t+6\Delta t (it will become clear why I picked 6\Delta t shortly).
Now, let's decrease the time step, one step at a time, until we come infinitely close to \vec{r_t}.
This is the result. When this occurs, the vectors \vec{r_i} and \vec{r_f} become almost indistinguishable. (Notice in this picture that as \Delta t decreases [6-times total] the vector \vec{r_f} moves to the
left and becomes darker in color.)
Additionally, the vector \Delta \vec{r} becomes tangent to the red curve. The red curve is the path that the particle follows; therefore, the velocity vector lies tangent to the objects direction of motion. This is simply a graphical representation of what it means to take the limit of:
\vec{v}(t) = \frac{d \vec{r}(t)}{dt} = \dot{ \vec{r}} (t) = lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}
___Acceleration____________
As mentioned, the acceleration is the limit of the time rate of change of the velocity, or the second derivative of the time rate of change of the position vector:
\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2 \vec{r}}{dt^2}
To 'see' what's going on with the acceleration vector, we create what's called a
hodograph. A hodograph describes the
locus of points for the velocity vector centered about a common origin.
At each instant in time, the particle will have a velocity vector that is tangent to the path. This is shown by the
top curve, or "trajectory" in http://eom.springer.de/common_img/h047500a.gif diagram.
If we were to take each of these velocity vectors along the trajectory and give them a common origin, O, we would get a graph similar to the one on the
bottom of the figure. There would be a
path traced out through the collection of the tip of
each velocity vector.
Consider the velocity vector on the bottom figure of the hodograph labeled v. This can be any velocity vector at any time t. The velocity vector to its
right can be another velocity vector at a later time t+\Delta t. Just as in the case of the position vector, if we take the limit as \Delta t goes to zero, we find that the acceleration vector becomes
tangent to the velocity vector
on the hodograph.
Caution
The acceleration vector is tangent to the
hodograph and is generally not tangent to the path of motion. Why? The acceleration vector is directly affected by the change in the both the magnitude and direction of the velocity vector. If only the magnitude of the velocity vector were changing, then the acceleration would be tangent to the path of motion as well as to the hodograph; however, whenever the velocity vector changes direction it tends to swing the velocity vector towards the inside, or concave side of the path. As a result, the acceleration cannot remain tangent to the path.
*Note*
When we take the limit as \Delta t \rightarrow 0 we call this the
instantaneous velocity or the
instantaneous acceleration. When we have a
discretization of the data such that the velocity or accleration is found by using a
finite difference, we call that the
average velocity or the
average acceleration over the time interval \Delta t. Here is an example:
-Instantaneous velocity:
\vec{v}(t) = lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t+ \Delta t) -\vec{r}(t)}{\Delta t}=\frac{d\vec{r}}{dt}
-Discritized average velocity:
\vec{v}(t)_{avg} = \frac{\vec{r}(t+ \Delta t) -\vec{r}(t)}{\Delta t}=\frac{\Delta \vec{r}}{\Delta t}
Although the average velocity and acceleration are not as accurate as the instantaneous velocity and acceleration, don't underestimate its imporantance. In real world systems, the velocity and acceleration are measured by analog devices and converted into a digitial signal which is sent to the computer. An analog device outputs a continuous signal; however, mircoprocessors work in binary (1's and 0's). Therefore, the analog signal must first be converted into a digital signal the processor can use. This is done by an
analog-to-digital converter (A/D). During this process, the continuous analog signal becomes discritized (or
quantized) to finite time step on the x-axis and a finite
amplitude zone on the y-axis. How small the time step is depends on the quality of the sampling rate of the A/D. The result is that the computer
has to calculate the
average velocity and acceleration of real world systems and use it as an
approximation of the instantaneous velocity or acceleration.
Here is an example of how an analog signal (the gray curve) becomes converted into a digital signal (the red curve). The dashed lines represent the discretization of time on the x-axis, and the voltage on the y-axis. During any time interval \Delta t, the red curve approximates the gray curve as being constant over said time span.