A boat travels at a speed of vBW relative to the water in a river of width D . The speed at whic

step 1 All right, we've got a boat traveling in the river. So the boat travels at a speed V -sub B -W.

A boat travels at a speed of vBW relative to the water in a...

A boat travels at a speed of $v_{\mathrm{BW}}$ relative to the water in a river of width $D .$ The speed at which the water is flowing is $v_{\mathrm{W}}$ a) Prove that the time required to cross the river to a point exactly opposite the starting point and then to return is $T_{1}=2 D / \sqrt{v_{B W}^{2}-v_{W}^{2}}$ b) Also prove that the time for the boat to travel a distance $D$ downstream and then return is $T_{1}=2 D v_{\mathrm{B}} /\left(v_{\mathrm{BW}}^{2}-v_{\mathrm{w}}^{2}\right)$

A boat travels at a speed of $v_{\mathrm{BW}}$ relative to the water in a river of width $D .$ The speed at which the water is flowing is $v_{\mathrm{W}}$
a) Prove that the time required to cross the river to a point
exactly opposite the starting point and then to return is $T_{1}=2 D / \sqrt{v_{B W}^{2}-v_{W}^{2}}$
b) Also prove that the time for the boat to travel a distance $D$ downstream and then return is $T_{1}=2 D v_{\mathrm{B}} /\left(v_{\mathrm{BW}}^{2}-v_{\mathrm{w}}^{2}\right)$

Key Concepts

Frame of Reference and Resultant Velocity

Choosing an appropriate frame of reference, such as the stationary ground versus the moving water, is critical in solving relative motion problems. The concept of resultant velocity, which is the vector sum of the boat's velocity relative to the water and the water's current velocity, allows one to determine the actual path and effective speed of the boat for each segment of its trip.

Kinematic Time Calculation

Kinematic time calculation involves determining the time taken to cover a given distance by dividing that distance by the effective speed in the direction of travel. When the motion is decomposed into components, this principle is applied separately to each direction, and the overall time for the journey is derived from these individual calculations.

Relative Motion

Relative motion refers to the analysis of an object's motion by considering the relative velocities between different frames of reference. In scenarios involving a moving medium, such as a river, the velocity of the object (like a boat) relative to the stationary ground is obtained by vectorially adding its own velocity relative to the medium and the velocity of the medium itself.

Vector Decomposition

Vector decomposition is the process of breaking a velocity or force vector into perpendicular components. In problems involving crossing a river, the boat's velocity is decomposed into a component perpendicular to the river bank (which affects the crossing distance) and a component parallel to the river bank (due to the current), allowing for the correct calculation of effective speeds in each direction.

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