(d) average velocity of the trip 4. (a) Calculate the distance and average speed in $$ m \cdot \mathrm{~s}^{-1} $$ from A to B if the duration of the journey is 2 minutes (b) Aso calculate the displacement and average velocity from A to B . (Hint circumference of circle $$ 2 \pi R $$ or $$ \pi \mathrm{O} $$ )

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To solve the problem of calculating the **distance**, **average speed**, **displacement**, and **average velocity** for a trip from point **A** to point **B**, we'll make use of the provided hint regarding the circumference of a circle.

**Assumptions:**
1. The trip from **A** to **B** follows a semicircular path.
2. The radius of the circular path is denoted as $$ R $$.
3. The duration of the journey is **2 minutes**.

If there are specific values for the radius $$ R $$ or other parameters, please provide them to refine the calculations. For now, we'll proceed with general formulas.

### 4. (a) Calculate the Distance and Average Speed

**Distance Traveled ($$ d $$):**
Since the trip follows a semicircular path, the distance traveled is half the circumference of the circle.

$$
\text{Circumference of a full circle} = 2\pi R
$$

$$
\text{Distance from } A \text{ to } B = \frac{1}{2} \times 2\pi R = \pi R
$$

**Average Speed ($$ v_{\text{avg}} $$):**
Average speed is the total distance traveled divided by the total time taken.

$$
v_{\text{avg}} = \frac{d}{t}
$$

Given:
- $$ t = 2 \text{ minutes} = 120 \text{ seconds} $$

$$
v_{\text{avg}} = \frac{\pi R}{120} \quad \text{(in meters per second)}
$$

### 4. (b) Calculate the Displacement and Average Velocity

**Displacement ($$ \Delta x $$):**
Displacement is the straight-line distance from the starting point **A** to the endpoint **B**. For a semicircular path, this is the diameter of the circle.

$$
\Delta x = 2R
$$

**Average Velocity ($$ \vec{v}_{\text{avg}} $$):**
Average velocity is the displacement divided by the time taken. It's a vector quantity, meaning it has both magnitude and direction.

$$
|\vec{v}_{\text{avg}}| = \frac{\Delta x}{t} = \frac{2R}{120} = \frac{R}{60} \quad \text{(in meters per second)}
$$

*Direction:* The direction of the average velocity vector is the straight line from **A** to **B**.

### Summary of Formulas

- **Distance ($$ d $$)**: $$ \pi R $$
- **Average Speed ($$ v_{\text{avg}} $$)**: $$ \frac{\pi R}{120} \, \text{m/s} $$
- **Displacement ($$ \Delta x $$)**: $$ 2R $$
- **Average Velocity ($$ \vec{v}_{\text{avg}} $$)**: $$ \frac{R}{60} \, \text{m/s} $$ towards **B** from **A**

### Example Calculation

If you provide a specific radius $$ R $$, we can plug in the numbers to get numerical values. For instance, let's assume $$ R = 50 $$ meters:

- **Distance**: $$ d = \pi \times 50 \approx 157.08 $$ meters
- **Average Speed**: $$ v_{\text{avg}} = \frac{157.08}{120} \approx 1.31 $$ m/s
- **Displacement**: $$ \Delta x = 2 \times 50 = 100 $$ meters
- **Average Velocity**: $$ |\vec{v}_{\text{avg}}| = \frac{50}{60} \approx 0.83 $$ m/s towards **B** from **A**

Please provide additional details or specific values if you need a more precise calculation!
To find the distance and average speed from A to B: - **Distance**: $$ \pi R $$ meters - **Average Speed**: $$ \frac{\pi R}{120} $$ m/s For displacement and average velocity: - **Displacement**: $$ 2R $$ meters - **Average Velocity**: $$ \frac{R}{60} $$ m/s towards B from A

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