# A boat travelling at top speed upstream moves at 15 km per hour. When it travels downstream, again at top speed, it moves at 25 km per hour. What is the boat top speed in still water?

## Please help me this question is linear systems, and I think involves substitute. I would appreciate so much a quick detailed response. Thank you

May 18, 2018

Let speed of boat in still water be$= {v}_{B}$
Let speed of stream be$= {v}_{S}$

Given

Upstream $15 = {v}_{B} - {v}_{S}$ ...........(1)
Downstream $25 = {v}_{B} + {v}_{S}$ ...........(2)

Adding (1) and (2) to eliminate ${v}_{S}$ we get

$15 + 25 = {v}_{B} - {v}_{S} + {v}_{B} + {v}_{S}$
$\implies 2 {v}_{B} = 40$
$\implies {v}_{B} = \frac{40}{2} = 20 \setminus k m \cdot {h}^{-} 1$

Jul 13, 2018

$20 \text{ km per hour}$

#### Explanation:

$\textcolor{b l u e}{\text{Some thinking before we start.}}$

Going up stream the boat is fighting the current. So $\underline{\text{on land}}$ it looks as though it is travelling slowly.

On the other hand, going down stream it is travelling with the current so $\left(\text{on land}\right)$ it looks as though it is travelling faster.

Its is all about 'relativity'. The boat is travelling at a constant speed in relation to just the water.
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$\textcolor{w h i t e}{\text{d}}$
$\textcolor{b l u e}{\text{Answering the question}}$

Let the flow speed of the water in relation to land be $w$
Let the boat speed in relation to water be $b$

Going upstream the boat is fighting the current of the water.
So

Boat upstream related to land is $15 \frac{k m}{h} = \left[b - w\right] \frac{k m}{h}$

Boat downstream related to land is $25 \frac{k m}{h} = \left[b + w\right] \frac{k m}{h}$

For convenience drop the units of measurement for now

$15 = b - w \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots E q u a t i o n \left(1\right)$
$25 = b + w \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left(2\right)$

$E q n \left(1\right) + E q n \left(2\right)$ cancels out $w$ giving

$40 = 2 b$

divide both sides by 2

$b = 20 \text{ km per hour}$

Thus the boat speed in relation to water is $20 \frac{k m}{h}$