Starting from one shore, you swim east across a narrow river to the other shore. The river is 19.0m wide. As you swim, the river current moves you north up the river a distance of 12.0m . Draw a diagram representing the situation. What is your resultant displacement? Express your answer in components, and then determine the magnitude

To solve this problem, let's first understand the situation with a diagram and then calculate the resultant displacement.

### Diagram Explanation:

1. **River Width (East Direction):** This is the width of the river you swim across, which is 19.0 meters. This displacement is directly eastward.
2. **River Current Displacement (North Direction):** As you swim eastward, the river's current moves you north by a distance of 12.0 meters.

We can represent these displacements as vectors:
- The eastward displacement is \( \vec{D}_\text{east} = 19.0 \, \text{m} \) along the x-axis.
- The northward displacement is \( \vec{D}_\text{north} = 12.0 \, \text{m} \) along the y-axis.

### Diagram:

```
Y (North)
^
|   (12.0 m)
|  
|
|----------------> X (East)
     (19.0 m)
```

### Resultant Displacement:

To find the resultant displacement, we use vector addition. The components of the displacement vector are:

- \( \Delta x = 19.0 \, \text{m} \) (east)
- \( \Delta y = 12.0 \, \text{m} \) (north)

The resultant displacement vector \( \vec{R} \) can be written in component form as:

\[
\vec{R} = (19.0 \, \text{m}, 12.0 \, \text{m})
\]

#### Magnitude of the Resultant Displacement:

To find the magnitude \( R \) of the resultant displacement vector, we use the Pythagorean theorem:

\[
R = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]

Substituting the values:

\[
R = \sqrt{(19.0)^2 + (12.0)^2}
\]

\[
R = \sqrt{361 + 144}
\]

\[
R = \sqrt{505}
\]

\[
R \approx 22.47 \, \text{m}
\]

### Result:

- **Resultant Displacement in Components:** \( (19.0 \, \text{m}, 12.0 \, \text{m}) \)
- **Magnitude of Resultant Displacement:** \( 22.47 \, \text{m} \)

Would you like a more detailed explanation or have any questions?

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Here are some related questions you might find interesting:

1. What is the angle of the resultant displacement with respect to the east direction?
2. If the river current were stronger, how would that affect the resultant displacement?
3. How would you represent the problem if the river had a current moving southwest instead of north?
4. Can you find the time taken to swim across the river if your swimming speed is 2 m/s in still water?
5. How would the diagram change if the current moved you south instead of north?

**Tip:** When dealing with vectors, always consider both their magnitude and direction to fully understand their impact on the overall motion.

Learn how to calculate resultant displacement when swimming across a river with a 19.0m width, affected by a 12.0m northward river current. Includes vector addition and application of the Pythagorean theorem. Suitable for high school students.

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