Question

Two towns A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man cycling with a speed of 20 km h¯¹ in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. What is the period T of the bus service and with what speed (assumed constant) do the buses ply on the road?

Answer 1

Given, the speed of the cycle is 20  km/h , the time in which the bus passes the cyclist in his direction of motion is 18 min and in opposite to his direction is 6 min .

Let v b be the speed of the bus between towns A and B and v c be the speed of the cyclist.

The relative speed of the bus is,

v= v b − v c

The distance covered by the bus is,

d=( v b − v c )×18 min =( v b − v c )×18 min× 1 s 60 min =( v b − v c )× 18 60 …… (1)

Let T be the time in which every bus leaves from the bus stop.

The distance covered by the bus is,

D= v b × T 60 …… (2)

Equate equation (1) and (2).

( v b − v c )× 18 60 = v b × T 60 …… (3)

The relative velocity of the bus in opposite the motion of the cycle is v b + v c . In this case the distance covered by the bus is,

D=( v b + v c )×6 min …… (4)

Equate the equation (2) and (4).

( v b + v c )×6 min= v b × T 60 ( v b + v c )× 6 60 = v b × T 60 …… (5)

Divide equation (3) by (5).

( v b − v c )× 18 60 ( v b + v c )× 6 60 = v b × T 60 v b × T 60

Substitute the required values in the above expression.

( v b −20  km/h )× 18 60 ( v b +20  km/h )× 6 60 =1 v b =40  km/h

Substitute the required values in equation (5).

( 40  km/h +20  km/h )× 6 60 =40  km/h × T 60 T=9 min

Thus, the T of the bus service is 9 min, and the speed at which the buses ply on the road is 40 km/h.