2.3 Accelerated motion in one dimension

Motion in one dimension is the basic component of all motion. A general three dimensional motion is equivalent to a system of three linear motion along three axes of a rectangular coordinate system. Thus, study of one dimensional accelerated motion forms the building block for studying accelerated motion in general. The basic defining diferential equations for velocity and acceleration retain the form in terms of displacement and position, except that they consist of displacement or position component along a particular direction. In terms of represntation, position vector r is replaced with x i or y j or z k in accordance with the direction of motion considered.

The defining equations of velocity and acceleration, in terms of position, for one dimensional motion are (say in x-direction) :

$$\begin{array}{l}\mathbf{v}=\frac{đx}{đt}\mathbf{i}\end{array}$$

and

$$\begin{array}{l}\mathbf{a}=\frac{{đ}^{2}x}{đt^2}\mathbf{i}\end{array}$$

Only possible change of direction in one dimensional motion is reversal of motion. Hence, we can define velocity and acceleration in a particular direction, say x - direction, with equivalent scalar system, in which positive and negative values of scalar quantities defining motion represent the two possible direction.

The corresponding scalar form of the defining equations of velocity and acceleration for one dimensional motion are :

$$\begin{array}{l}v=\frac{đx}{đt}\end{array}$$

and

$$\begin{array}{l}a=\frac{{đ}^{2}x}{đt^2}\end{array}$$

It must be clearly understood that the scalar forms are completely equivalent to vector forms. In the scalar form, the sign of various quantities describing motion serves to represent direction.

Nature of acceleration in one dimensional motion

One dimensional motion results from the action of net external force that applies along the direction of motion. It is a requirement for motion to be in one dimension. In case, force and velocity are at certain angle to each other, then there is sideway deflection of the object and the resulting motion is no more in one dimension.

If velocity and force are in the same direction, then magnitude of velocity increases; If velocity and force are in the opposite direction, then magnitude of velocity decreases.

The valid combination (i and ii)and invalid combination (iii) of velocity and acceleration for one dimensional motion are shown in the figure.