## Boats & Streams

**Basic Terms used in the chapter:**

1) Stream: The moving water of river is called as stream.

2) Still Water: The water which is not moving is known as still water.

3) Upstream: In water, the direction against the stream is known as upstream.

4) Downstream: In water, the direction along the stream is known as downstream.

5) In this chapter the problems are based on the effect of water current on the movement of boats. There are two important phenomena which form the basis for this chapter - upstream (against the water current) & downstream (along with the water current).

If there's no mention of stream speed in the question, assume it to be the speed of boat in still water.

6) Problems similar to boats & streams may also occur which include -

a) Cyclist & Wind (where cyclist is analogous to boat while wind is analogous to stream)

b) Swimmer & Stream (where swimmer is analogous to boat)

**Important Formulae:**

1) If the speed of boat in still water is 'x' km/hr and the speed of stream is 'y' km/hr the

Downstream speed (S

_{d}) = (x + y) km/hr

2) While going downstream, boat moves in the direction of flow of river. So, water stream increases the speed of boat. Hence, speeds are added while going downstream.

Upstream Speed (S

_{u}) = (x – y) km/hr

3) While going upstream, boat moves in the direction opposite to that of flow of river. So, water stream decreases the speed of boat due to opposite flow. Hence, speeds are subtracted while going upstream.

In still water,

Speed of boat (x) = ( | 1 | ) x [Downstream speed(S_{d}) + Upstream speed(S_{u})] |

2 |

If you just remember this concept, you won't need to remember and recall the above formula while solving the problems.

4) If you are given speed of the boat upstream and downstream and are required to find out the speed of the current, the following concept is very useful:

Downstream Speed – Upstream Speed = 2 (Speed of the current)

x + y – (x – y) = 2y.

Halving this would give you the speed of the current.

i.e. Speed of stream (y) = ( | 1 | ) x [Downstream speed(S_{d}) – Upstream speed(S_{u})] |

2 |

5) If a boat moves at 'x' km/hr speed and covers the same distance up and down in a stream of speed 'y' km/hr, then average speed of boat is calculated by,

Average Speed = | Downstream Speed x Upstream Speed | = | (x + y) (x – y) | km/hr |

Speed in still water | x |

6) If a boat takes time 't' hours more going upstream than to move downstream for the same distance, then the distance is given by,

Distance = | [(x^{2} – y^{2}) (t)] | km |

(2y) |

7) If a boat moves to a certain distance downstream in 't

_{1}' hours & returns the same distance upstream in time 't

_{2}' hours, then

Speed of boat in still water = y | (t_{2} + t_{1}) | km/hr |

(t_{2} – t_{1}) |

8) If a boat takes 't' hours to move a certain place & come back again, then

Distance between palces = | [t (x_{2} – y_{2})] | x km |

2 |

**The problems from this chapter can be divided into four types. Once you know how to deal with these 4 types of problems, the chapter should be an easy one for you.**

**Question Variety:**

Type 1 : Speed of boat in still water and speed of stream are given separately. Find the time taken by boat to go upstream & downstream

**Examples:**

**Q 1.**If a boat travels with a speed of 10 km/hr in still water and the speed of stream is 5 km/hr, what would be the time taken by boat to go 60 km downstream?

a. 2 hrs

b. 4 hrs

c. 6 hrs

d. 8 hrs

View solution

Correct answer :(b)**Hint: In downstream, water stream increases the speed of boat as they both are along the same directions. Hence, both the speeds are added.**

Thus, Downstream speed (S_{d}) = (x + y) km/hr --------------(1)

Time = | Distance | ------------------------------ (2) |

Speed |

= 10 + 5 = 15 km/hr

We have,

Time = | Distance |

Speed |

Time taken by a boat = | 60 | = 4 hours |

15 |

**Q 2.**A man rows a boat at 8 km upstream in 2 hours and 2 km downstream in 40 minutes. How long will he take to reach 7 km in still water?

a. 2 hrs

b. 4 hrs 15 min

c. 6 hrs

d. 7 hrs 40 min

View solution

Correct answer: (a)**Hint: In still water, the speed of boat is given by,**

Speed of boat (x) = ( | 1 | ) x [Downstream speed(S_{d}) + Upstream speed(S_{u})] |

2 |

We need to know the values of upstream & downstream speeds.

Upstream Speed = | Distance | = | 8 | = 4 km/hr |

Time | 2 |

Downstream Speed = | Distance | = | 2 | km/min = | 2 | x 60 = 3 km/hr |

Time | 40 | 40 |

Hence, speed of boat ( x) = | 1 | [3 + 4] = | 7 | = 3.5 km/hr |

2 | 2 |

Thus, the time required to reach the distance of 7 km = | Distance Covered | = | 7 | = 2 hrs |

Speed of boat | 3.5 |