A particle moves on a straight line in such a way it covers 1st half distance with speed 3m/s and next half distance in two equal time intervals with speed 4.5m/s and 7.5m/s respectively. Find average speed of the particle.

let total distance of path is S ...

for first half , time taken if t

speed = distance/time

3 = S/2t

t = S/6      .............1

for second half , time taken is T ...

for first  T/2 it moves with 4.5

speed = distance/time

4.5 = 2x/T

 x = 4.5T/2   ...........2

for second T/2 it moves with 7.5

7.5 = 2y/T

y = 7.5T/2       ..........3

we have , x+y = S/2so

(7.5+4.5)T = S/2

T = S/24        ...............4

total time taken during its motion is T+t

 T_total = T+t = S/24+S/6 = 5S/24

total distance = S

average speed = total distance/total time = S/5S/24 = 24/5 = 4.8m/s