Consider a particle moving along x-axis as shown in figure. Its distance from the origin O is described by the coordinate x, which varies with time. At a time t 1 , the particle is at point P, where its coordinate is x 1 , and at time t 2 it is at point Q, where its coordinate is x 2 . The displacement during the time interval from t 1 to t 2 is the vector from P to Q: the x-component of this vector is (x 2 – x 1 ) and all other components are zero.

It is convenient to represent the quantity x 2 – x 1 , then change in x, by means of a notation using the Greek letter Δ (capital delta) to designate a change in any quantity. Thus we write Δ x = x 2 – x 1 in which Δ x is not a product but is to be interpreted as a single symbol representing the change in the quantity x. Similarly, we denote the time interval from t 1 to t 2 as Δ t = t 2 – t 1 .

The average velocity of the particle is defined as the ratio of the displacement Δ x to the time interval Δ t. We represent average velocity by the letter v with a bar to signify average value. Thus

(i) A particle moves half the time of its journey with u. The rest of half time it moves with two velocities V 1 and V 2 such that half the distance it covers with V 1 and the other half with V 2 . Find the net average velocity.

Assume straight line motion.