Find the displacement of a farmer who walks 80m [E] then 60m [E 25° S]."









To calculate the farmer's **displacement**, we need to add the two vectors representing the farmer's movement. The first vector is \( 80 \, \text{m} \) due east, and the second is \( 60 \, \text{m} \) at an angle of \( 25^\circ \) south of east. We can break the second vector into its **east** and **south** components, then sum the components to get the total displacement.

### Step 1: Break down the second vector (60 m [E 25° S])
The second vector can be split into:
- **East component** \( = 60 \cos(25^\circ) \)
- **South component** \( = 60 \sin(25^\circ) \)

Calculating the components:
\[
\text{East component} = 60 \cos(25^\circ) \approx 60 \times 0.9063 = 54.38 \, \text{m}
\]
\[
\text{South component} = 60 \sin(25^\circ) \approx 60 \times 0.4226 = 25.36 \, \text{m}
\]

### Step 2: Sum the components
- The total **eastward displacement** is:
\[
80 \, \text{m} + 54.38 \, \text{m} = 134.38 \, \text{m}
\]
- The total **southward displacement** is:
\[
25.36 \, \text{m}
\]

### Step 3: Find the magnitude of the resultant displacement
The magnitude of the total displacement is the resultant of these two components:
\[
\text{Displacement} = \sqrt{(134.38)^2 + (25.36)^2}
\]
Let me calculate this.

\[
\text{Displacement} \approx \sqrt{134.38^2 + 25.36^2} = \sqrt{18060.8 + 643.3} = \sqrt{18704.1} \approx 136.7 \, \text{m}
\]

### Step 4: Find the direction of the displacement
The direction of the displacement can be found using the arctangent of the ratio of the south and east components:
\[
\theta = \tan^{-1}\left(\frac{\text{South component}}{\text{East component}}\right) = \tan^{-1}\left(\frac{25.36}{134.38}\right)
\]
Let me calculate this as well.

\[
\theta \approx \tan^{-1}(0.1887) \approx 10.7^\circ
\]

So, the farmer's displacement is approximately **136.7 m at 10.7° south of east**.

---

Would you like further details or have any questions?

### Related questions:
1. How would the result change if the second vector had a different angle?
2. What is the difference between displacement and distance in this context?
3. How can we calculate the displacement if the path is curved instead of straight?
4. What would the displacement be if both vectors were due east?
5. How can we compute displacement in three dimensions?

### Tip:
Always break vectors into components when they involve angles for easier calculation of displacement.

Find the displacement of a farmer who walks 80m [E] then 60m [E 25° S].

Learn how to calculate the displacement of a farmer walking 80m east and 60m at 25° south of east. This problem involves vector addition, trigonometry, and displacement calculations. Suitable for students in Grades 9-11.

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