dunno how to solve this...?

Question

A man who can swim with velocity u reative to water takes time t1 to cross river in shortest time.It takes t2 for the same distance in the direction of river flow.if velocity of river is v,then u/v is? plz give the whole process.

Answer 1

First let’s define some variables,
Let the distance, d, be the distance across the river.
Let u be the velocity of the swimmer, and v be the velocity of the river current.
Times t_1 and t_2 are, respectively, the minimum time it takes the swimmer to cross the river the time it takes the swimmer to cover the same distance, D, as before, only in the direction of the river flow.
The distance, D, the swimmer covers during the quickest crossing is not equal to d, the distance across the river since the swimmer is swimming at a diagonal to this distance. The swimmer ends up moving a distance y down river while swimming a distance d across the river, making distance D equal to the square root of y^2 + d^2.
Since the distance y equals the velocity of the river * the time it took the swimmer to cross (t_1), y = t_1 * v, thus making D = sqrt ((v*t_1)^2 + d^2), but for simplicity, I will just call it D.

I have already just partially covered part of the method solving the problem, by so thoroughly defining the distance D.
The shortest time the swimmer can cross the river in is if he exerts all of his velocity in the direction pointed straight across the river (perpendicular to the river flow). Since perpendicular vectors are independent, the river flow adds a perpendicular component of velocity to him while at the same time, not taking anything away from his river crossing speed....meaning that the swimmer will move down river as well for an overall distance of D.
t_1 = "horizontal" change in distance / "horizontal" velocity
t_1 = d / t_1
[I put "horizontal" in quotations in order to try to indicate that it is not really horizontal, just in the picture I drew to solve the problem, I just mean to indicate that the change in distance in one component of the displacement vector divided by the velocity in that same direction yields the time it took to make the crossing]

When the swimmer swims in the direction of the river flow, now the two velocities (v and u) are no longer independent, they act in the same direction and thus add together. The resulting velocity = v + u. In order to cover the same distance D with this new, higher, velocity, it takes a time of t_2.
t_2 = Distance covered / velocity
t_2 = D / (v + u)

Solving for u and v with the two equations we now have for the times we get,
u = d / t_1
v = (D / t_2) - u

So,
u / v = (d / t_1) / ((D / t_2) - u)
This can probably be simplified to look "prettier", but I will leave that to you.

The thing about these types of questions is that they usually want you to express the answer in terms of specific variables (usually the variables given at the beginning of the question)...but since you did not specify, I left it in this form.

Answer 2

Ok, so the swimmer goes across kinda in a crossways line that takes t1 at V1. The distance the swimmer goes down stream isn't important because it is independent of the distance it goes across the river. The angle which the swimmer goes in is dependent on the speed down the river to the speed across (sin and cos components). U/v is the tangent of the angle the swimmer goes. I'm guessing from what you are saying here, T2 is the time it takes to go straight across the river against the current. If that is the case, U/V is the Arctan(T1/T2).

Answer 3

I'm not a math guy but it sounds like u want to no distance verses velocity, some can travel(distance) with no resistance(velosity)at a specific speed, wish I could meet that person, bt i'm going for chinise food, I no its going to take me ten minutes and so do they, how? because of distance and velocity

Answer 4

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