## Cottage Dwellers

### Task number: 1990

The residents of an inlet have their cottages built near a river with width *L*. When neighbours want to meet, they use boats to travel.

Mr. Smith owns a boat that sails on still water with constant speed *v*_{b} (speed refers to the magnitude of velocity). He decides to visit Mr. Jones with his neighbour, Mr. Brown.

Determine how long the journey will take if you know the following:

Mr. Smith will first go to Mr. Brown, who lives on the same side of the river, away at distance x against the current. Then, they will both sail to Mr. Jones, who lives on the other side of the river in a cottage exactly opposite Mr. Brown's cottage.

Assume the velocity of the current is constant and has a magnitude *v*_{r}.

Solve for values: *L* = 200 m, *v*_{b} = 2 m·s^{-1}, *v*_{c} = 1 m·s^{-1}, *x* = 500 m.

#### Given values

*L*= 200 mriver width *v*_{b}= 2 m·s^{−1}boat speed on still water *v*_{c}= 1 m·s^{−1}current speed *x*= 500 mdistance between the houses of Mr. Smith and Mr. Brown *t*= ? (s)duration of the journey #### Hint 1: Time

*t*_{1}from Mr. Smith to Mr. BrownDetermine the time

*t*_{1}it takes for Mr. Smith to sail to Mr. Brown.You know the distance of the journey. What is the velocity

*v*_{1}with which Mr. Smith sails to Mr. Brown?Draw a free body diagram describing the situation.

#### Hint 2 : Time

*t*_{2}from Mr. Brown to Mr. JonesDetermine the time

*t*_{2}it takes for Mr. Smith and Mr. Brown to sail to Mr. Jones.Once again, you know the distance of the journey. In what direction do they sail? Can they head straight to Mr. Jones's cottage?

What is the speed

*v*_{2}with which they sail? Draw a free body diagram.#### Hint 3: Total duration

*t*of the journeyWhat holds for the total duration of the journey

*t*? Can you express it using previous calculations?#### OVERALL SOLUTION

First we will determine the time

*t*_{1}it will take Mr. Smith to sail to Mr. Brown and then the time*t*_{2}it will take them to sail to Mr. Jones.\[t=t_1+t_2.\]

**Time***t*_{1}from Mr. Smith to Mr. BrownFigure 1:

Speed of the boat

\[v_\mathrm{1}=v_\mathrm{l}-v_\mathrm{c}.\]*v*_{1}in relation to the shore is, if it sails against the current, equal to the difference between the speed of the boat on still water*v*_{b}and the speed of the current*v*_{c}:The boat must travel distance

\[t_\mathrm{1}=\frac{x}{v_\mathrm{l}-v_\mathrm{c}}.\]*x*with this speed. It will take time*t*_{1}:**Time***t*_{2}from Mr. Brown to Mr. JonesIf Mr. Smith and Mr. Brown want to land at Mr. Jones's cottage, they cannot head straight to it (the current would carry them away). Instead, they need to sail in a manner depicted below:

Figure 2:

Net speed:

\[v_\mathrm{2}=\sqrt{v_\mathrm{b}^{2}-v_\mathrm{c}^{2}}.\]Time

\[t_\mathrm{2}=\frac{L}{v_\mathrm{2}}=\frac{L}{\sqrt{v_\mathrm{b}^{2}-v_\mathrm{c}^{2}}}.\]*t*_{2}:We have determined time

*t*_{1}it took Mr. Smith to sail to Mr. Brown and then time*t*_{2}it took them to sail to Mr. Jones.We obtain the total duration of the journey

\[ t=t_\mathrm{1}+t_\mathrm{2}=\frac{x}{v_\mathrm{b}-v_\mathrm{c}}+\frac{L}{\sqrt{v_\mathrm{b}^{2}-v_\mathrm{c}^{2}}}.\]*t*by adding both times:Numerically:

\[t=\left(\frac{500}{2-1}+\frac{200}{\sqrt{3}}\right)\,\mathrm{s}=615\,\mathrm{s}.\]#### Answer

Mr. Smith and Mr. Brown will get to Mr. Jones in time t:

\[ t=\frac{x}{v_\mathrm{l}-v_\mathrm{r}}+\frac{L}{\sqrt{v_\mathrm{l}^{2}-v_\mathrm{r}^{2}}}.\]Numerically:

\[t=615\,\mathrm{s}.\]