Swimmer's problem or River-boat problem: I.cl V1 v= velocity...

Swimmer's problem or River-boat problem: I.cl $V_{1 v}=$ velocity of water w.r.t ground Vs, $w=$ velocity of swimmer w.r.t. water $V_{s}=$ velocity of swimmer w.r.l. ground Now, $A C=V_{s}^{\prime} \times$ crossing time $t$ $\Rightarrow \quad(A B+B C)=\left(V_{s, w}^{\prime}+V_{w}^{\prime}\right)$ $\Rightarrow \quad(d \hat{j}+x \hat{i})=\left[\left(-V_{s, w}^{\prime} \sin \theta\right) \hat{i}+\left(V_{s, 18} \cos \theta\right) \hat{j}+i_{w} \hat{i}\right] t$ $\left.\Rightarrow \quad(x \hat{i}+d \hat{j})=\left(V_{v}-V_{s, 1 \mathrm{~F}} \sin \theta\right) t \hat{i}+\left(\ddot_{t, v} \cos \theta\right) t \hat{j}\right)$ i.c. $\quad x=\left(V_{w}-V_{A, v} \sin \theta\right) t$ and $d=\left(V_{s, v} \cos \theta\right) t \quad \Rightarrow \quad i-\frac{d}{\left(\sigma_{n, w}{\cos \theta)}\right.} \quad \ldots(3)$

Swimmer's problem or River-boat problem:
I.cl $V_{1 v}=$ velocity of water w.r.t ground Vs, $w=$ velocity of swimmer w.r.t. water $V_{s}=$ velocity of swimmer w.r.l. ground
Now, $A C=V_{s}^{\prime} \times$ crossing time $t$
$\Rightarrow \quad(A B+B C)=\left(V_{s, w}^{\prime}+V_{w}^{\prime}\right)$
$\Rightarrow \quad(d \hat{j}+x \hat{i})=\left[\left(-V_{s, w}^{\prime} \sin \theta\right) \hat{i}+\left(V_{s, 18} \cos \theta\right) \hat{j}+i_{w} \hat{i}\right] t$
$\left.\Rightarrow \quad(x \hat{i}+d \hat{j})=\left(V_{v}-V_{s, 1 \mathrm{~F}} \sin \theta\right) t \hat{i}+\left(\ddot_{t, v} \cos \theta\right) t \hat{j}\right)$
i.c. $\quad x=\left(V_{w}-V_{A, v} \sin \theta\right) t$
and $d=\left(V_{s, v} \cos \theta\right) t \quad \Rightarrow \quad i-\frac{d}{\left(\sigma_{n, w}{\cos \theta)}\right.} \quad \ldots(3)$

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Key Concepts

Relative Velocity

This concept involves understanding how the motion of an object is perceived differently depending on the frame of reference. In problems like the swimmer’s or river-boat problem, the velocity of the object relative to the ground is a vector sum of its velocity relative to the moving medium (water) and the medium’s own velocity relative to the ground.

Vector Addition

Vector addition is used to combine the individual velocity components, taking into account both magnitude and direction. This technique is critical for obtaining the resultant velocity vector that describes the actual path of the moving object in the reference frame of the ground.

Vector Decomposition

Decomposing vectors into their orthogonal components (usually along the horizontal and vertical axes) allows one to analyze the motion separately in each direction. This method employs trigonometric functions such as sine and cosine to resolve the vectors, simplifying the calculations involved in relative motion problems.

Kinematics (Displacement-Time Relations)

Kinematics connects displacement, velocity, and time. In the context of a swimmer or boat, the time it takes to cross a river or move from one point to another is determined by the effective speed in that direction, which is calculated after resolving the various vector components.

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