9.3 Crosses the river in shortest interval of time

  Motion in One Dimension

    1.0 Introduction
    2.0 Kinematic variables
    3.0 Motion in one dimension
    4.0 Derivation of the kinematics equation
    5.0 Vertical motion under gravity
    6.0 Analysis of motion through graph
    7.0 Relative motion
    8.0 Simultaneous motion of two bodies
    9.0 River boat problem
    10.0 Aircraft-wind problem
    11.0 Rain problem

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9.3 Crosses the river in shortest interval of time

Consider a river of width $d$ is flowing from left to right with velocity $v_R$.

For crossing the river in the shortest interval of time, all the power of the boat should be utilized in crossing the river.

So, the boat should travel along $y$ axis or perpendicular to the river as shown in the figure.

We know,
$${\overrightarrow v _B} = {\overrightarrow v _{BR}} + {\overrightarrow v _R}$$
${\overrightarrow v _{BR}}$ is along $y$ axis: So, it is responsible for motion along $y$ axis i.e. $AB$.
${\overrightarrow v _{R}}$ is along $x$ axis: So, it is responsible for motion along $x$ axis i.e. $BC$.

Resultant of motion along $AB$ and $BC$ is $AC$.

Observer on shore will observe that boat is going along $AC$.

$v_{BR}$ is only responsible for crossing the river.

So, time taken to cross the river is, $$t = \frac{d}{{{v_{BR}}}}$$
$\left( {t = \frac{d}{{{v_{BR}}}}} \right)$ will be the shortest time to cross the river.

Note:

The boat was traveling from $A$ to $B$ but actually, it was going along $AC$. So, instead of reaching point $B$, it reached point $C$.

So, $BC$ is termed as drift $(d_f)$.

Mathematically drift is given as, $${d_f} = {v_R}t$$or$${d_f} = \left( {\frac{{{v_R}}}{{{v_{BR}}}}} \right)d$$