(a) Assume a bus line with N stops, where distance between stops is S . Free flo
ID: 1765182 • Letter: #
Question
(a) Assume a bus line with N stops, where distance between stops is S. Free flow speed of thebus is v, with acceleration and deceleration, a. P passengers per stop are boarding and alighting and time needed for a passenger to board or alight is T seconds. What is the average speed of a bus?
(b) Following on part (a), assume the origin of passengers are spread uniformly along the bus route. The destination of all passengers is the last stop. The walking speed of passengers is u. Determine the average travel time of passengers.
NOTE: If you need to assume other variables and parameters to solve this problem clearly state that in your report and explain the reason.
Explanation / Answer
( a )
Average Speed is given by : Total distance travelled / Total Time taken
1. Total distance travelled: There are N stops with a distance S in between each of them. Hence, total distance travelled by the bus would be equal to : ( N - 1 ) * S
2. Total time taken: There are 3 parts of journey.
a. Time for boarding/alighting of passengers at N stops = N * P * T
b. Time in accleration/deceleration = ( v / a ) + ( v / a )............where v is free flow speed and a is the acceleration/deceleration rate. This happend in between every two stops.
c. Time in between when there is free flow speed = ( S - ( v * v / ( a ) ) / v..........Free flow speed is maintained in between each stop except for the distance it is acclerating/decelerating. This happens in between every two stops.
Hence. Total time taken = ( NPT ) + ( N - 1 ) [ ( 2v/a ) + ( S/v - v/a ) ] = ( NPT ) + ( (v/a) + (S/v) ) * ( N - 1 )
Hence, Average Speed of the bus = [ ( N - 1 ) * S ] / [ ( NPT ) + ( N - 1 ) * ( (v/a) + (S/v) ]
( b )
If passengers are spread uniformly along the bus route and every passenger's destination is last stop, with walking speed of passengers being u.
Travel time for first passenger at first stop to the last stop would be = ( N - 1 ) * S / u
Travel time for second passenger at second stop to the last stop would be = ( N - 2 ) * S / u
And similarly for remaining passengers till Nth stop.
Addition of all these times with a ration to N would be the average travel time.
Hence, Average travel time = ( S / u * N ) * ( N * ( N - 1 ) / 2 ) = [ ( N - 1 ) * S ] / [ ( 2 * u ) ]
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