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Question: A person rows a boat in water with a speed \(4m/s\), water in the river is flowing at a speed of \(2...

A person rows a boat in water with a speed 4m/s4m/s, water in the river is flowing at a speed of 2m/s2m/s. If the person rows the boat perpendicular to the direction of the flow, find the resultant velocity of the boat and time taken by the boat to cross the river, if the width of the river is 400  m400\;m.

Explanation

Solution

Since the boat is moving on a river which is flowing, then we can say that, the velocity of the river acts as an hindrance for the movement of the boat. For the boat to cross the river, clearly it must overcome the velocity of the river.

Formula used: vr=vb2+vw2v_{r}=\sqrt{v_{b}^{2}+v_{w}^{2}} and vel=disttimevel=\dfrac{dist}{time}

Complete step by step answer:
Let us consider that boat starts from a point OO as shown in the figure. Let the velocity of the boat be vb=4m/sv_{b}=4m/s and the velocity of the vw=2m/sv_{w}=2m/s. Also given that the width of the river is d=400  md=400\;m.
Let us say that the boat from OO reaches the perpendicular point AA in the absence of the vwv_{w}. Due to the velocity of the river, vwv_{w}, clearly, this hinders the movement of the boat and thus the boat from OO reaches BB , instead of AA, which is inclined at an angle θ\theta with respect to O  AO\;A

Let vrv_{r} be the resultant vector. Then from the parallelogram law of vector addition, we can say that figure we can say that the resultant velocity vrv_{r}
Then the magnitude of the resultant vector is given as vr=vb2+vw2v_{r}=\sqrt{v_{b}^{2}+v_{w}^{2}}
    vr=42+22=16+4=20=25m/s\implies v_{r}=\sqrt{4^{2}+2^{2}}=\sqrt{16+4}=\sqrt{20}=2\sqrt5 m/s
Also we know that, vel=disttimevel=\dfrac{dist}{time}
Given that the width of the river is d=400  md=400\;m and the velocity acting on the same axis is vbv_{b}
Then, t=4004=100st=\dfrac{400}{4}=100s

Thus the velocity of the resultant is vr=25v_{r}=2\sqrt5 and the time taken t=100st=100s

Note: This sum may seem very hard at first glance, but if one draws the diagram and labels the diagram correctly, it is easy to solve the question. The question uses only the basic parallelogram law of vector addition and speed formula.