Question

Between two points, a car accelerates uniformly at first, then moves with constant speed and finally retards uniformly to rest. If the ratio of time takes is `1:4:2` and the maximum speed is `70 km//h`, find the average speed over the whole journey.

Answer 1

The motion is in three phases, acceleration, constant velocity and then retardation. Let the time for three phase be `t_(0)`, `4t_(0)` and `2t_(0)`. The maximum velocity will be at the end of the first phase and the beginning of the third phase.

A to B: (constant acceleration)
`d_(1)=((0+v_(m))/2) t_(0)` `("Displacement" = "Average velocity" xx "Time")`
`B` to `C`: (constant velocity)
`d_(2)=v_(m)xx4t_(0)=4v_(m)t_(0)`
`C` to `D`: (constant retardation)
`d_(3)=((v_(m)+0)/2)xx2t_(0)=v_(m)t_(0)`
Average speed`=("Total distance")/("Total time")`
`=(d_(1)+d_(2)+d_(3))/(t_(0)+4t_(0)+2t_(0))`
`=((v_(m)t_(0))/2+4v_(m)t_(0)+v_(m)t_(0)+v_(m)t_(0))/(7t_(0))`
`=(11 v_(m))/14=11/14xx70=55 km//h`