Vector Calculus Swimming Problem

[BigFlorida]

Homework Statement

[/B]
A swimmer located at point A needs to reach a point B 20 meters downstream on the opposite bank of a 10 meter wide river. The river flows horizontally at a rate of 0.5 meters/second, and the swimmer has a constant speed of 0.25 meters/second.

Set up the vector equations needed to determine the velocity vector of the swimmer, then determine the velocity vector of the swimmer.

Homework Equations

Length: ||s|| = sqrt( s₁² + ... + s_n²)

The Attempt at a Solution

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I do not know why this question has given me so much trouble, as all the others have went relatively well. I know that:
*speed is the length of the velocity vector,

*the swimmer's x-component of velocity is (I think) <0.25cos(theta) , 0 > (this is the component for the current, but I know something is not right when I try to take the magnitude and it does not come out as 0.5, but I do not know what else to do to it.)

*the swimmer's y-component of velocity is (I think) <0, 0.25sin(theta)>

*Together, these give the swimmer a velocity, relative to the ground, <0.25cos(theta), 0.25sin(theta)> which seems to be correct when the length is evaluated.

Also, I used the length and width of the river to find the angle. Is all of this wrong? Is there something blatantly obvious that I am missing?

I am just at a loss and have spent hours on this question. Any help would be very much appreciated.

 

[Mark44]

Homework Statement

[/B]
A swimmer located at point A needs to reach a point B 20 meters downstream on the opposite bank of a 10 meter wide river. The river flows horizontally at a rate of 0.5 meters/second, and the swimmer has a constant speed of 0.25 meters/second.

Set up the vector equations needed to determine the velocity vector of the swimmer, then determine the velocity vector of the swimmer.

Homework Equations

Length: ||s|| = sqrt( s₁² + ... + s_n²)

The Attempt at a Solution

[/B]
I do not know why this question has given me so much trouble, as all the others have went relatively well. I know that:
*speed is the length of the velocity vector,

*the swimmer's x-component of velocity is (I think) <0.25cos(theta) , 0 > (this is the component for the current, but I know something is not right when I try to take the magnitude and it does not come out as 0.5, but I do not know what else to do to it.)

*the swimmer's y-component of velocity is (I think) <0, 0.25sin(theta)>

*Together, these give the swimmer a velocity, relative to the ground, <0.25cos(theta), 0.25sin(theta)> which seems to be correct when the length is evaluated.

Also, I used the length and width of the river to find the angle. Is all of this wrong? Is there something blatantly obvious that I am missing?

I am just at a loss and have spent hours on this question. Any help would be very much appreciated.

The swimmer needs to swim in the direction of a point across the river somewhere between point B and a point directly opposite A. The river's velocity is twice what the swimmer's velocity would be in still water, so that the swimmer's velocity in the river will be considerably more than if he were swimming in a lake. The resultant of the two vectors gives you the velocity vector of the swimmer.

Draw a sketch of a vector representing the swimmer's direction and velocity, and the river's direction and velocity.

 

[HallsofIvy]

It makes no sense to talk about "x and y components" until you have specified which direction is "x" and which "y". Suppose we take "x" to be "down the river" and "y" to be "across the river". Then we can write the river's velocity vector as < 0.5, 0>. Write the swimmer's velocity vector (in still water), in meters per minute, as <v_x, v_y>.

[Some text removed by a moderator as too much help]