A body starts from rest and travels a distance S with uniform acceleration, then moves uniformly a distance 2S and finally comes to rest after moving further 5S under uniform retardation. Find the ratio of average velocity to maximum velocity :

Explanation

Graphically:Area of (V-t) curve represent displacement

`S=1/2V_"max"t_1` or `t_1=(2S)/V_"max"`
`2S=V_"max"t_2` or `t_2=(2S)/V_"max"`
`5S=1/2 V_"max"t_3` or `t_3=(10S)/V_"max"`
`V_(av)="Total displacement"/"Total time"`, `V_"av"=(S+2S+5S)/("2S"/V_"max"+"2S"/V_"max"+"10S"/V_"max")`
`V_(av)/V_"max"=(8S)/(14S)=4/7`
`V_(av)/V_(max)="Total displacement"/(2("total displacement during acceleration and retardaton")+("Displacement During uniform velocity "))`
`therefore V_"av"/V_"max"="8S"/(2(S+5S)+2S)=8/14=4/7`