A truck is moving at a constant speed of 50km/h on a straight road which terminates on a wall. A fly starts moving with a constant speed of 100km/h from the wall towards the truck when the truck is at a distance 25km from the wall. Fly reaches the truck and then turns back towards the wall and then turns back towards the wall and then on reaching the wall it again turns towards the truck and so on. it makes several trips between the truck and the wall, before the truck just reaches the wall.
(a) What is the total distance travelled by the fly during this period?
(b) how many trips the fly makes between the truck and the wall?

Explanation

(a) time taken by the truck to reach the wall
`=("Distance")/("speed")=(25km)/(50km//h)=(1)/(2)h`,
Hence with a speed of 100km/hr, fly will cover 50 km in `(1)/(2)` hr.
(b) let us assume that at a certain instant fly is at the wall and truck is at a distance `x_(1)` from the wall. fly and truck are moving towards each other with speeds 100km/h and 50km/h, respectively. hence relative speed becomes 100+50=150km/h. time taken by the fly to reach the truck can be written as follows:
`t_(1)=(x_(1))/(150)`
Distance travelled by the fly to reach the truck
`x=100t_(1)=100xx(x_(1))/(150)=(2)/(3)x_(1)`
Distance travelled by truck in time interval
`x"=50xxt_(1)=50xx(x_(1))/(150)-(x_(1))/(3)`
Further time taken by the fly to reach back the wall
`t_(2)=(x)/(100)=(2x_(1))/(3xx100)=(x_(1))/(150)`
Further distance travelled by truck in time `t_(2)`
`x=50xxt_(2)=50xx(x_(1))/(150)=(x_(1))/(3)`
Distance between truck and the wall after trip of the fly is complete can be written as follows:
`x=x_(1)-x-x=x_(1)-(x_(1))/(3)-(x_(1))/(3)=(x_(1))/(3)`
`rArr x_(2)=x_(1)//3`
Using the above result we can write the following Distance between the truck and the wall at the beginning of `1^("st")` trip=20km
Distance between the truck and the wall at the beginning of `2^("nd")` trip`=((1)/(3))xx20` km
Distance between the truck and the wall at the beginning of `3^("rd")` trip`=((1)/(3))^(2)xx20` km
Similarly
Distance between the truck and the wall at the beginning
of `n^("th")` trip`=((1)/(3))^(n-1)xx20` km
Distance between the truck and the wall will reduce to zero only when n approaches infinity. Hence, the fly will theoretically make infinite trips between truck and the wall before truck touches the wall.