An object is said to be in motion if its position changes with time. The position of the object can be specified with reference to a conveniently chosen origin. For motion in a straight line, the position to the right of the origin is taken as positive and to the left as negative.
The path length is defined as the total length of the path traversed by an object.
Displacement is the change in position: \(\Delta x=x_{2}-x_{1}\). The path length is greater or equal to the magnitude of the displacement between the same points.
An object is said to be in uniform motion in a straight line if its displacement is equal in equal intervals of time. Otherwise, the motion is said to be non-uniform.
Average velocity is the displacement divided by the time interval in which the displacement occurs:
\(\bar{v}=\frac{\Delta x}{\Delta t}\)
On an \(x\)-t graph, the average velocity over a time interval is the slope of the line connecting the initial and final positions corresponding to that interval.
Average Speed is the ratio of total path length traversed and the corresponding time interval. The average speed of an object is greater or equal to the magnitude of the average velocity over a given time interval.
Instantaneous velocity or simply velocity is defined as the limit of the average velocity as the time interval \(\Delta t\) becomes infinitesimally small:

\(v=\lim _{\Delta t \rightarrow 0} \bar{v}=\lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{\mathrm{d} x}{\mathrm{~d} t}\)
The velocity at a particular instant is equal to the slope of the tangent drawn on position-time graph at that instant.

Average acceleration is the change in velocity divided by the time interval during which the change occurs:
\(\bar{a}=\frac{\Delta v}{\Delta t}\)
Instantaneous acceleration is defined as the limit of the average acceleration as the time interval \(\Delta t\) goes to zero:
\(a=\lim _{\Delta t \rightarrow 0} \bar{a}=\lim _{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t}=\frac{\mathrm{d} v}{\mathrm{~d} t}\)
The acceleration of an object at a particular time is the slope of the velocity-time graph at that instant of time. For uniform motion, acceleration is zero and the \(x-t\) graph is a straight line inclined to the time axis and the \(v-t\) graph is a straight line parallel to the time axis. For motion with uniform acceleration, \(x\) – \(t\) graph is a parabola while the \(v\) – \(t\) graph is a straight line inclined to the time axis.
The area under the velocity-time curve between times \(t_{1}\) and \(t_{2}\) is equal to the displacement of the object during that interval of time.
For objects in uniformly accelerated rectilinear motion, the five quantities, displacement \(x\), time taken \(t\), initial velocity \(v_{0}\), final velocity \(v\) and acceleration \(a\) are related by a set of simple equations called kinematic equations of motion:

\(
\begin{aligned}
& \quad v=v_{0}+a t \\
& \quad x=v_{0} t+\frac{1}{2} a t^{2} \\
& \quad v^{2}=v_{0}^{2}+2 a x
\end{aligned}
\)

if the position of the object at time \(t=0\) is 0 . If the particle starts at \(x=x_{0}, x\) in above equations is replaced by \(\left(x-x_{0}\right)\).

[Note: Here the bar over the symbol for velocity is a standard notation used to indicate an average quantity.]

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