A particle moves in a straight line from `A` to `B`
(a) for the first half of distance with speed `v_(1)` and the next half of distance with speed `v_(2)`.
(b) for the first one-third distance with speed `v_(1)` and the next two-third distance with speed `v_(2)`.
(c ) for the first one-fourth distance with speed `v_(0)`, the next half of distance with speed `2v_(0)` and the last one-fourth distance with speed `3v_(0)`.
Find the average speed in each case.

Explanation

(a) Let the total distance be d.

`A` to `C`: `t_(1)=(d//2)/v_(1)=d/(2v_(1))`
`C` to `B: t_(2)=(d//2)/v_(2)=d/(2v_(2))`
Average speed `=d/(d/(2v_(1))+d/(2v_(2)))=(2v_(1)v_(2))/(v_(1)+v_(2))`
(b)
`CB` is double of `AC`.
`A` to `C: t_(1)=(d//3)/v_(1)=d/(3v_(1))`
`C` to `B: t_(2)=(2d//3)/v_(2)=(2d)/(3v_(2))`
Average speed `=d/(d/(3v_(1))+(2d)/(3v_(2)))=(3v_(1)v_(2))/(2v_(1)+v_(2))`
(c )
`A` to `C: t_(1)=(d//4)/v_(0)=d/(4v_(0))`
`C` to `D: t_(2)=(d//2)/(2v_(0))=d/(4v_(0))`
`D` to `B: t_(3)=(d//4)/(3v_(0))=d/(12v_(0))`
Total time `=d/(4v_(0))+d/(4v_(0))+d/(12v_(0))`
`(d(3+3+1))/(12v_(0))=(7d)/(12v_(0))`
Average speed `=d/(7d//12v_(0))=(12v_(0))/7`