**Instantaneous Speed **

**Instantaneous Speed**

The velocity of a particle at some instant of time or at some point on its path is termed as its instantaneous velocity. We know that velocity of the particle is the ratio of displacement to the time interval, however, an instant of time is having zero duration and consequently, a body can undergo no displacement precisely at an instant, then how can we define instantaneous velocity. In Physics, an instant means a very small-time interval which almost approaches zero.

as Δt → 0, the average velocity becomes instantaneous velocity. So,

*i.e., instantaneous velocity is the rate of change of position or displacement. In other words, we can say that instantaneous velocity at a point (or at an instant) may be defined as the average velocity of an extremely small displacement (or time interval) which includes the point (or instant).*

As the duration of instant is very small and in this duration displacement is also very small, so we can say for an instant distance and displacement are having same values, so instantaneous speed is nothing but the magnitude of the instantaneous velocity.

where ds is the distance travelled by the particle in small time interval dt.

♦The slope of the tangent drawn to the position-time graph at any point gives instantaneous velocity while numerical value of slope gives instantaneous speed.

♦If the velocity is constant, then speed would be constant, but if speed is constant then velocity may not be constant as direction may change.

**Acceleration**

When the velocity of a particle is changing with time then the particle is said to have some acceleration. From the figure, the particle is having velocity

Then the average acceleration for time interval t_{2} – t_{1} is defined as,

While instantaneous acceleration is defined as,

It is a common misconception that direction of acceleration is same as that of the direction of velocity, so be careful!

The direction of acceleration is the same as that of change in velocity vector

The above discussed definitions of displacement, distance, velocity, acceleration etc are valid for all types of motion, whether the particle is moving along a straight line path or curved path.

♦When the absolute magnitude of the velocity of a particle is decreasing, acceleration is termed as deceleration or retardation and the body is said to be decelerated or retarded. It is absolutely wrong to say that –ve acceleration is deceleration.

♦ If the velocity of a particle is constant, then the acceleration would be zero but if speed is constant then acceleration may or may not be zero.

Representation of Rectilinear Motion