How Should a Swimmer Cross a Rapid River to Minimize Downstream Travel?

[a1231234]

Homework Statement

1)A swimmer wishes to across a swift, straight river of width d. If the speed of the swimmer in still water is u and the speed of the water is v (> u), what is the direction along which the swimmer should proceed such that the downstream distance he has traveled when he reaches the opposite bank is the smallest possible? What is the minimum downstream distance? What is the corresponding time required?

2)As an extension of the last example, if the swimmer wants to cross the swift with a minimum time, find the minimum time and the position that he reaches in the opposite bank.

Homework Equations

The Attempt at a Solution

Ans1)The direction along which the swimmer should proceed should be from Q to C
sin∅=u/v AB=d cot ∅ =(d√ 1-(u/v)^2 )/ (u/v) = d√( v^2-u^2)/u
Time taken : t =d / u cos ∅ = d /(1 - (u^2/v^2) )

Problem 1)What confuses me is that t =d / u cos ∅ , isn't that the net velocity of the swimmer equals to the red-line vector ?? Why is it possible to calculate time of travel using part of the velocity ( u cos ∅) ?

Ans2) The minimum time of travel can be obtained if the swimmer directs in the direction of . The magnitude of is contributed completely in the direction of crossing the swift. It is the fastest way. Hence the minimum time t = d/u. The distance downstream s = vt = (vd)/u.

Problem 2) Regardless of the magnitude of u , the total velocity must be affected by the water speed v , which is equals to the red-line vector . why not t= (d/sin∅)/red-line velocity ? For the downstream distance , why is it possible to calculate by multiplying time and part of the total velocity (v ) ?

These confusion has long been troubling me ,so can anyone give me a help ??Thanks very much

 

[anathema]

Dear forum post,

Thank you for your interesting question regarding a swimmer crossing a swift river. I would like to provide some insights and explanations to help clarify your confusions.

Firstly, in order to find the direction along which the swimmer should proceed, we need to consider the relative velocities of the swimmer and the water. In this case, the swimmer's velocity in still water is u and the water's velocity is v. The swimmer should proceed in the direction where the relative velocity between the swimmer and the water is minimized. This is because in this direction, the swimmer will experience the least resistance from the water and therefore will be able to cross the river with the least effort.

To calculate the downstream distance and time required, we use the concept of relative velocity. The downstream distance is the distance between the starting point and the point where the swimmer reaches the opposite bank. This distance is equal to the product of the swimmer's speed in still water (u) and the time taken to cross the river (t). The time taken to cross the river is calculated using the relative velocity between the swimmer and the water, which can be expressed as v-u. Therefore, the downstream distance can be written as d = u(t) or t = d/u.

To answer your first question, the reason why we use the component of the swimmer's velocity (u cos ∅) to calculate the time is because this component represents the part of the swimmer's velocity that is in the direction of the river's flow. This is the component that is relevant to the swimmer's motion across the river. The other component (u sin ∅) is perpendicular to the river's flow and does not contribute to the swimmer's motion across the river.

For your second question, the reason why we use the total velocity (v) to calculate the downstream distance is because the total velocity represents the resultant velocity of the swimmer and the water. This is the velocity that determines the swimmer's motion across the river. The component of the total velocity (v sin ∅) represents the part of the velocity that is perpendicular to the river's flow and does not contribute to the swimmer's motion across the river. Therefore, we only need to consider the component of the total velocity that is in the direction of the river's flow, which is v cos ∅, to calculate the downstream distance.

I hope this helps