Ayana and Ashton are in canoes along a stretch of the DesPlaines River. They typically paddle at the same pace and can travel at the same speed as one another (relative to the water). But Ayana heads upstream while Ashton heads downstream. We will define the positive direction as downstream. An observer on the shore observes that Ayana has a velocity of -1.03 m/s and Ashton has a velocity of +2.81 m/s. What is the river velocity? That is, what is the velocity of the water relative to the shore?

To determine the velocity of the river (i.e., the water velocity relative to the shore), we can utilize the given velocities of Ayana and Ashton, as well as the fact that their paddling speed relative to the water is the same.

### Problem Breakdown:
- **Ayana's velocity relative to the shore** is given as \(-1.03 \, \text{m/s}\) (since she is paddling upstream).
- **Ashton's velocity relative to the shore** is given as \(+2.81 \, \text{m/s}\) (since he is paddling downstream).
- Their speed relative to the water (paddling speed) is the same, which means:
  - Ayana's velocity relative to the shore is the result of her paddling speed relative to the water subtracted by the river's velocity.
  - Ashton's velocity relative to the shore is the result of his paddling speed relative to the water added to the river's velocity.

Let’s define:
- \(v_{\text{river}}\) = the velocity of the river relative to the shore.
- \(v_{\text{paddle}}\) = their paddling speed relative to the water (the same for both Ayana and Ashton).

We can write two equations based on the observations:
1. \(v_{\text{Ayana}} = v_{\text{paddle}} - v_{\text{river}}\), where \(v_{\text{Ayana}} = -1.03 \, \text{m/s}\)
2. \(v_{\text{Ashton}} = v_{\text{paddle}} + v_{\text{river}}\), where \(v_{\text{Ashton}} = 2.81 \, \text{m/s}\)

### Step-by-Step Solution:

1. Add the two equations to eliminate \(v_{\text{paddle}}\):
   \[
   (-1.03) + 2.81 = (v_{\text{paddle}} - v_{\text{river}}) + (v_{\text{paddle}} + v_{\text{river}})
   \]
   Simplifying:
   \[
   1.78 = 2v_{\text{paddle}}
   \]
   \[
   v_{\text{paddle}} = \frac{1.78}{2} = 0.89 \, \text{m/s}
   \]

2. Now, substitute \(v_{\text{paddle}}\) into one of the original equations to solve for \(v_{\text{river}}\). Using the equation for Ayana:
   \[
   -1.03 = 0.89 - v_{\text{river}}
   \]
   Solving for \(v_{\text{river}}\):
   \[
   v_{\text{river}} = 0.89 + 1.03 = 1.92 \, \text{m/s}
   \]

### Final Answer:
The velocity of the river relative to the shore is \(1.92 \, \text{m/s}\) downstream.

Would you like further details, or do you have any questions about the solution?

Here are 5 related questions:
1. How would the velocities change if both Ayana and Ashton were paddling at different speeds?
2. What if Ayana paddled faster than Ashton; how would that affect the equations?
3. How do upstream and downstream velocities typically affect the travel times of boats?
4. Can we determine the time it takes for either paddler to cover a certain distance if we know their paddling speed relative to the water?
5. How would the problem change if the river's current varied along its path?

**Tip:** When analyzing relative velocities, always clearly define the reference frame (such as "relative to the shore" or "relative to the water"). This helps in setting up the correct equations.

Discover how to calculate the river velocity using relative motion with a step-by-step guide. This problem involves Ayana paddling upstream and Ashton paddling downstream along the DesPlaines River. Using their observed velocities of -1.03 m/s and 2.81 m/s respectively, the solution breaks down key concepts of relative velocity and linear equations. Perfect for students in Grades 9-12.

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