# 1.17 Understanding motion

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Our success or failure in understanding motion largely depends on our ability to identify motion according to a certain scheme of classification.

The discussion of different attributes of motion in previous modules has led us to the study of motion from the point of view of a general consideration to a simplified consideration such as uniform or rectilinear motion. The time is now ripe to recapitulate and highlight important results - particularly where distinctions are to be made.

For convenience, we shall refer general motion as the one that involves non-linear, two/ three dimensional motion. The simplified motion, on the other hand, shall refer motion that involves one dimensional, rectilinear and uniform motion.

Consideration of scalar quantities like distance and speed are same for “general” as well as “simplified” cases. We need to score similarities or differences for vector quantities to complete our understanding up to this point. It is relevant here to point out that most of these aspects have already been dealt in detail in previous modules. As such, we shall limit our discussion on main points/ results and shall generally not use figures and details.

## Similarities and differences

Similarity / Difference 1 : In general, the magnitude of displacement is not equal to distance.

$\begin{array}{l}|\Delta \mathbf{r}|\le s\end{array}$

For rectilinear motion (one dimensional case) also, displacement is not equal to distance as motion may involve reversal of direction along a line.

$\begin{array}{l}|\Delta x|\le s\end{array}$

For uniform motion (unidirectional motion),

$\begin{array}{l}|\Delta x|=s\end{array}$

Similarity / Difference 2 : The change in the magnitude of position vector is not equal to the magnitude of change in position vector except for uniform motion i.e motion with constant velocity.

For two/three dimensional motion,

$\begin{array}{l}\Delta r\ne |\Delta \mathbf{r}|\end{array}$

For one dimensional motion,

$\begin{array}{l}\Delta x\ne |\Delta x|\end{array}$

For uniform motion (unidirectional),

$\begin{array}{l}\Delta x=|\Delta x|\end{array}$

Similarity / Difference 3 : In all cases, we can draw a distance – time or speed – time plot. The area under speed – time plot equals distance (s).

$\begin{array}{l}s=\int vdt\end{array}$

Similarity / Difference 4 : There is an ordered sequence of differentiation with respect to time that gives motional attributes of higher order. For example first differentiation of position vector or displacement yields velocity. We shall come to know subsequently that differentiation of velocity, in turn, with respect to time yields acceleration. Differentiation, therefore, is a tool to get values for higher order attributes.

These differentiations are defining relations for the attributes of motion and hence applicable in all cases irrespective of the dimensions of motion or nature of velocity (constant or variable).

For two or three dimensional motion,

$\begin{array}{l}\mathbf{v}=\frac{d\mathbf{r}}{dt}\end{array}$

For one dimensional motion,

$\begin{array}{l}v=\frac{dx}{dt}\end{array}$

Similarity / Difference 5 : Just like differentiation, there is an ordered sequence of integration that gives motional attributes of lower attributes. Since these integrations are based on basic/ defining differential equations, the integration is applicable in all cases irrespective of the dimensions of motion or nature of velocity (constant or variable).

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