A boat moves relative to water with a velocity half of the river flow velocity. If the angle from the direction of flow at which the boat must move relative to stream direction to minimize drift is $\frac{2 π}{n}$ , then find $n$ .

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Solution

Let the velocity of boat is $v$ and making an angle $θ$ with the vertical and the width of the river is $d$ . Then the velocity of river is $2 v$ in horizontal direction.

The resultant velocity of boat is given as,

$v_{x} = 2 v - v sin θ$

$v_{y} = v cos θ$

The required time to cross the river is given as,

$t = \frac{d}{v cos θ}$

The drift is given as,

$x = v_{x} \times t$

$x = 2 v - v sin θ \times \frac{d}{v cos θ}$

For drift to be minimum,

$\frac{d x}{d θ} = 0$

$2 v tan θ - v sec θ = 0$

$θ = 30^{\circ}$

The angle made with the direction of river flow is given as,

$θ_{1} = 90^{\circ} + 30^{\circ}$

$θ_{1} = 120^{\circ}$

The value of $n$ is given as,

$\frac{2 π}{n} = 120^{\circ}$

$n = 3$

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A boat moves relative to water with a velocity half of the river flow velocity. If the angle from the direction of flow at which the boat must move relative to stream direction to minimize drift is 2πn , then find n .

A boat moves relative to water with a velocity half of the river flow velocity. If the angle from the direction of flow at which the boat must move relative to stream direction to minimize drift is $\frac{2 π}{n}$ , then find $n$ .