A car is moving at a constant speed of 40 km/h along a straight road which heads towards a large vertical wall and makes a sharp `90^0` turn by the side of the wall. A fly flyingat constant speed of 100 km/h, starts from the wall towrds the car t an instant when the car is 20 km away,flies until it reaches the glasspane of the car and returns to teh wll at teh same speed. It continues to fly between the car and teh wall time the car makes the `90^0` turn. a. What is the total distance the fly has travelled during the period?b. How many trips has it made between the car and the wall?

Explanation

a. The time takenby the car to cover 20 km before the turnis `(20 km)/(40 km/h)=1/2h`. The fly moves at a constant speed of 100 km/h during this time. Hence the total distance covred by it is `100 km/hxx1/2h=50 km`.
b. Suppose the car is at a distance dx away (at A) wqhen the fly is at teh wall (at O). The fly hits the glsspane at B, taking a time t. Then
b. Suppose the car is at a distasnce x away (at A) when the fly is at the wall (at O). The fly hits the glasspane at B, taking a time t. Then
`AB=(40 km/h)t`
and `OB=(100 km/h)t`
Thus, `x=AB+OB`
`=(140 km/h)t`
or, `t=x/(140 km/h) or OB = 5/7x`
. The fly returns to teh wall and during this period the car moves the distance BC. The time taken by the fly in this return path is
`((5x/7)/(100 km/h)=x/(140k/h)`
Thus, `BC=(40x)k/140 =2/7x`
or, `OC=OB-BC=3/7x`
IF at teh beginning opf the round trip (wall to the car and back) the car is a distance x away, it is `3/7x` away when the next trip again starts.
Distance of the car at the beginning of the 1st trip=20 km.
Distance of the car at the begining of the 2nd trip
`=3/7xx20 km`
Distance of the car at the beginning of the 3rd trip `(3/7)^2xx20km`,
Distance of the car at the beginning of the 4th trip
`=(3/7)^2xx20 km`
Distance of teh car at the begining of the nth trip
`=(3/7)^(n-1)xx20 km`
Trips will go on till the car reaches the turn the is the distance resuce of zero. This will be the case when n be comes infinity. Hence the fly makes an infinite number of tris before the car takes the turn.