We have studied in school physics that time rate of change of displacement is known as velocity. Its direction is along the direction of displacement. So if d is the total displacement of the body in time t, then its average velocity during the interval t is defined as

equation_velocity_02

Average velocity does not tell us about the motion between A and B. the path may be straight or curved and the motion may be steady or variable. For example it a squash ball comes back to its starting point after bouncing off the wall several times. Its total displacement is zero and so also is its average velocity.

In such cases the motion is described by the instantaneous velocity.

In order to understand the concept of instantaneous velocity, consider a body moving along a path ABC in xy plane. At any time t, let the body be at point A Fig. 3.2(b). its position is given by position vector r1. After a short time interval ∆t following the instant t, the body reaches the point B which is described by the position vector r2. The displacement of the body during this short time interval is given by

∆d = r2 – r1

The notation ∆ (delta) is used to represent a very small change.

The instantaneous velocity at a point A, can be found by making ∆t smaller and smaller. In this case ∆d will also become smaller and point B will approach A. if we continue this process, letting B approach A, thus, allowing ∆t and ∆d to decrease but never disappear completely, the ratio ∆d/∆t approaches a definite limiting value which is the instantaneous velocity. Although ∆t and ∆d become extremely small in this process, yet their ratio is not necessarily a small quantity. Moreover, while decreasing the displacement vector, ∆d approaches a limiting direction along the tangent at A. therefore,

The instantaneous velocity is defined as the limiting value of ∆d/∆t as the time interval ∆t, following the time, approaches zero.

Using the mathematical language, the definition of instantaneous velocity vins is expressed as

equation_velocity_01

Read as limiting value of ∆d/∆t as ∆t approaches zero.

If the instantaneous velocity does not change, the body is said to be moving with uniform velocity.

For your information

Typical speeds

Speed, ms-1           Motion
300 000 000   Light, radio waves. X-rays. Microwaves (in vacuum)

210 000   Earth-Sun travel around the galaxy

29 600   Earth around the Sun

  1 000   Moon around the Earth SR-71

     980   reconnaissance jet

     333   Sound (in air)

     267   commercial jet airliner

       62   commercial automobile (max.)

37   falcon in a dive

29   Running cheetah

10   100-metres dash (max.)

  9   porpoise swimming

  5   Flying bee

  4   human running

  2   human swimming

     0.01   Walking ant