A swimmer can swim in still water with speed 5 km/hr. What is the ratio of time taken by him to swim across a river in shortest time to that over shortest distance? River is flowing with speed 3 km/hr.
Answer:
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Answer:
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Given:
Speed of the river
Speed of the swimmer
To find:
The ratio of time taken by the swimmer to cross a river in the shortest time to that in the shortest distance.
Solution:
Step 1
If we consider the diagram for the journey for the shortest travelling time.
Let O be the starting point from which the swimmer should start swimming towards B in order to cross the river in the shortest time.
But, due to the flow of the river, the swimmer reaches point P when, OP will take the shortest time to cross the river.
In Δ, we have,
The vector for the speed of the river
and, the vector for the speed of the swimmer
We know,
Speed is the distance traveled by an object in a given time period.
We have,
The time taken to cross the river in shortest time as and the distance with a speed of
Step 2
If we consider the diagram of the journey for minimum travelling distance.
Let, A be starting point from where the swimmer should start swimming in order to reach P, which is the shortest distance.
But due to the flow of river, there is a high probability that the swimmer might lose the point P.
Hence, he needs to swim apparently in the diagonally further distance In order to reach P.
Now,
In Δ, using Pythagoras theorem, we get
We have,
The vector for thee speed of the river and the vector for the speed of the swimmer
Hence, the swimmer need to swim with a speed of 4 km/h in the AB direction in the running river in order to reach P from where, AP is the shortest distance possible to cross the river.
We know,
We have, the time taken by the swimmer to travel apparently AP as the distance with a speed of .
Step 3
Now, the ratio of time taken by the swimmer to cross the river in the shortest time to that the shortest distance will be,
Final answer:
Hence, the ratio of time taken by the swimmer to cross the river in the shortest time to the shortest distance is 5 :4.