An elevator accelerates from rest at a constant rate `alpha` for time interval `t_(1)` and travels a distance `S_(1)` It them retards at a constant rate `beta` for time interval `t_(2)` and finally comes to rest after travelling a distance `S_(2)` during its retardation Show that:
`(S_(1))/(S_(2))=(t_(1))/(t_(2))=(beta)/(alpha)`.

Explanation

Velocity-time graph for the above situation is as follows:

Maximum velocity attained by the elevator is `v_(0)`. Slope of the v-t graph represents acceleration of the body, hence we can write the following relation:
`alpha=(v_(0))/(t_(1))` and `beta=(v_(0))/(t_(2))`
`rArr (alpha)/(beta)=(t_(2))/(t_(1))`
Area under the v-t graph represents displacement, hence we can write the following,
`S_(1)=(1)/(2)v_(0)t_(1)` and `S_(2)=(1)/(2)v_(0)t_(2)`
Dividing the above two equation e get
`(S_(1))/(S_(2))=(t_(1))/(t_(2))=(beta)/(alpha)`