Rectilinear Motion - Instantaneous Speed

Instantaneous speed is simply the speed of an object at a single, specific moment in time (t).

To calculate instantaneous speed, we look at the average speed (Average Speed=ΔtDistance​) over a very, very short time interval (Δt). It is defined as the limiting value of the average speed as the time interval (Δt) approaches zero.

Instantaneous Speed = \[\operatorname*{lim}_{\Delta t\to0}\frac{\mathrm{Distance}}{\Delta t}\]

We can analyze motion visually using a Position-Time (x−t) Graph, where the object's position (x) is plotted on the vertical axis against time (t) on the horizontal axis.

The Golden Rule: Slope is Velocity!

The most important concept to remember is that the slope (or gradient) of the x−t graph gives you the object's velocity.

Velocity = Slope = \[\frac{\text{Change in Position}}{\text{Change in Time}}\]

Graph Type (Based on Figure 3.1)	Description	Key Features of the Graph	Real-Life Example
(a) At Rest	The object's position is not changing.	Horizontal straight line. Slope is zero.	A car is parked in a garage.
(b) Uniform Velocity (+)	The object moves at a constant speed in the positive (+x) direction.	Straight line with a constant positive slope. Velocity is constant and positive.	A train moving on a straight track under cruise control.
(c) Uniform Velocity (−)	The object moves at a constant speed in the negative (−x) direction.	Straight line with a constant negative slope. Velocity is constant and negative.	A ball rolling back towards its starting point at a steady rate.
(d) Oscillatory Motion	The object's position repeatedly changes direction (e.g., from +x to −x and back) over fixed time intervals.	Wavy or zig-zag pattern. The slope (velocity) alternates between positive and negative.	A mass oscillating on a spring or a pendulum swing.
(e) Non-Uniform Motion	The object's velocity is changing (it is accelerating or decelerating).	A curved line. The slope is not constant.	A car speeding up after a traffic light or slowing down to park.

For a curved x − t graph (non-uniform motion):

1. Average Velocity is the slope of the secant line (a straight line connecting two points, e.g., line AB or CD).
2. Instantaneous Velocity at any specific time (t₀) is the slope of the tangent line (a line that touches the curve at only one point, e.g., line PQ).

As the time interval around t0​ gets smaller (e.g., from line AB to CD, and so on), the secant line gets closer and closer to the tangent line (PQ). The slope of this tangent line gives the instantaneous velocity at that exact moment.