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Boats and Streams/Escalator

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Boats and streams should be ideally seen as just a logical extension of the motion in a straight Line, with distance being constant.
As we know, if the distance is constant then V α 1/T.

Basic Terminology

Downstream movement When the direction of the movement of a river and a boat is the same, their collective movement is known as the downstream movement. And the distance covered by boat is known as downstream distance.
If the speed of the River = R and the speed of the boat = B, then Downstream Speed = B + R
Upstream movement When the direction of the movement of the river and a boat is opposite, they are said to be in upstream movement. The distance covered in this case is known as upstream distance.
If the speed of the river = R and the speed of the boat = B, then upstream speed = B − R ?(Conventionally the speed of one boat is taken more the than speed of the river otherwise the boat would not be able to go back.)
Now, speed of boat = 1/2 (downstream speed + upstream speed) = 1/2 (B + R + B − R) = B
And speed of river = 1/2 (downstream speed − upstream speed) = 1/2 (B + R − B + R) = R
Hence, if downstream speed and upstream speed are given as 20 km/h and 10 km/h respestively, then the speed of the boat = 15 km/h and speed of the river = 5 km/h.
In most of the cases of boats and streams, the distances covered downstream and upstream are the same. In those cases, the ratio of the time taken becomes inverse of the ratio of the speeds.

Time taken downstream: Time taken upstream = upstream speed:downstream speed

The speed of the boat in still water is 6 km/h and the speed of the river is 1.2 km/h. Boat takes a total of 10 h to go to a place and come back. What is the total distance covered in the whole process?
Let us assume D is the distance.
Upstream Speed = 4.8 km/h
Downstream speed = 7.2 km/h
According to the question, D/4.8 + D/7.2 = 10
So, D = 28.8 km and hence, the total distance = 57.6 km
Alternatively, the ratio of downstream speed:upstream speed = 3:2
Ratio of the downstream time:upstream time = 2:3
The time taken in the downstream movement = 4 h and the time taken in the upstream movement = 6 h
So, the distance covered = 4 × 7.2 = 6 × 4.8 = 28.8 km
Hence, the total distance = 57.6 km

In the case of escalators, moving staircase works like an external agent as the river works for boats and streams. The speed of an escalator and the person will be added when the staircase is going up and the person walking up with it have the same direction of the movement.

Now if the direction of the movement of an escalator and the person are opposite, then the resultant speed (or, the relative speed) will be equal to the speed of the person − to the speed of an escalator.
A man can walk up in a moving escalator (upwards) in 30 seconds (s). The same man can walk down this moving ‘up’ escalator in 90 s. Assume that this walking speed is the same both upwards and downwards. How much time will he take to walk up the escalator when it is not moving?
Let us assume that the speed of the man = m steps/s and the speed of the escalator = e steps/s
Distance covered while going up = 30 m + 30 e
Distance covered while going down = 90 m − 90 e
Now, these two are equal.
So, 30 m + 30 e = 90 m − 90 e
Or, 60 m = 120 e Hence, 1 m = 2 e
So, the total length of escalator = 45 m
So, the time taken by the man to cover the whole escalator = Distance/Speed = 45 m/m = 45 s
Alternatively, Answer would be Harmonic Mean of the given time = Description: 9446.png seconds

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