Question

A particle cover each 1/3 of the total distance with speed v1,v2 and v3 reapectively. Find athe average speed of the particle?

Answer 1

Total distance S

Time to cover the first 1/3 of  S =  t1 = (S/3) / v1 = S/(3 v1)
Time to cover the second 1/3 of S = t2 = (S/3)/v2 = S/(3 v2)
Time to cover the last 1/3 of S = t3 = (S/3 ) / v3 = S/ (3v3)

Total time = t =  S/3 [ 1/v1 + 1/v2 + 1/v3 ]

Let v = average speed

v = S / t        So 1/v = t / S

1/ v = 1/3  [ 1/v1 + 1/v2 + 1/v3 ]

Answer 2

Total distance traveled = S

Time taken for the 3 parts of the journey = t1, t2 and t3 respectively

Speed during the 3 parts of the journey = v1, v2, v3, respectively

Distance covered in part of the journey = S/3

=> t1 = (S/3)/v1 = S/(3v1)

t2 = (S/3)/v2 = S/(3v2)

t3 = (S/3)/v3 = S/(3v3)

Total time taken = t = t1 + t2 + t3

= (S/3)(1/v1 + 1/v2 + 1/v3)

=Average speed = V(av) = S/t = S/[(S/3)(1/v1 + 1/v2 + 1/v3)]

= 1/[(1/3)(1/v1 + 1/v2 + 1/v3)]

= 1/[(v2 v3 + v3 v1 + v1v2)/{3(v1 v2 v3)}]

= (3 v1 v2 v3) / (v1 v2 + v2 v3 + v3 v1)