# Vector Calculus Swimming Problem

• BigFlorida
In summary, the problem involves a swimmer trying to reach a point on the opposite bank of a river while accounting for the river's horizontal flow. The swimmer has a constant speed and the river flows at a faster rate. To determine the swimmer's velocity vector, vector equations need to be set up and the resultant of the swimmer's velocity and the river's velocity must be taken into account. The direction of the x and y components must also be specified.
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 + ... + sn2)

## 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.

BigFlorida said:

## 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 + ... + sn2)

## 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.

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 <vx, vy>.

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

Last edited by a moderator:

## 1. What is Vector Calculus and how is it related to swimming?

Vector Calculus is a branch of mathematics that deals with vector fields and their derivatives, such as gradient, divergence, and curl. It is used to study the motion of objects in space, including swimming. By applying Vector Calculus, we can analyze the velocity and acceleration of a swimmer in different directions and determine the overall motion of the swimmer.

## 2. What are the key variables in the Vector Calculus Swimming Problem?

The key variables in the Vector Calculus Swimming Problem are the swimmer's position, velocity, and acceleration. These variables are represented by vectors in three-dimensional space and are used to determine the swimmer's trajectory and motion.

## 3. How is Vector Calculus used to optimize a swimmer's performance?

Vector Calculus can be used to optimize a swimmer's performance by analyzing their motion and identifying areas for improvement. For example, by calculating the swimmer's velocity and acceleration vectors, we can determine which direction they are most efficient in and suggest changes to their technique to improve overall performance.

## 4. What are some applications of Vector Calculus in swimming?

Aside from analyzing a swimmer's performance, Vector Calculus has many other applications in swimming. It can be used to study fluid dynamics and determine the flow of water around a swimmer's body, which can help in designing more efficient swimsuits. It is also used in analyzing the forces acting on a swimmer, such as drag and buoyancy, to improve their technique and reduce resistance in the water.

## 5. Is Vector Calculus necessary for competitive swimming?

While understanding Vector Calculus may not be essential for competitive swimming, it can provide valuable insights into a swimmer's performance and help them improve their technique. Many coaches and sports scientists use Vector Calculus to analyze and optimize a swimmer's motion, making it a useful tool for competitive swimmers who want to reach their full potential.

• Introductory Physics Homework Help
Replies
12
Views
2K
• Introductory Physics Homework Help
Replies
15
Views
3K
• Introductory Physics Homework Help
Replies
5
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
20
Views
665
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Classical Physics
Replies
33
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
3K
• General Math
Replies
11
Views
2K
• Classical Physics
Replies
24
Views
2K