A car traveled a semicircular path with a radius of 2 km in 3 minutes. Calculate its average speed and acceleration.

To solve this, we calculate the **average speed** and **average acceleration** of the car.

---

### Step 1: Average Speed
The formula for average speed is:

\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
\]

#### Total Distance:
Since the car travels a semicircular path, the total distance is the circumference of half a circle:

\[
\text{Total Distance} = \pi r = \pi (2) = 2\pi \, \text{km}
\]

#### Total Time:
The time given is 3 minutes. Convert this into hours:

\[
\text{Total Time} = \frac{3}{60} = 0.05 \, \text{hours}
\]

#### Average Speed:
Substitute the values into the formula:

\[
\text{Average Speed} = \frac{2\pi}{0.05} = 40\pi \, \text{km/hr} \approx 125.66 \, \text{km/hr}
\]

---

### Step 2: Average Acceleration
The formula for average acceleration is:

\[
\text{Average Acceleration} = \frac{\Delta v}{\Delta t}
\]

#### Initial and Final Velocities:
- The car starts at one point on the semicircle and ends at the opposite point.
- Since velocity is a vector, the **initial velocity** and **final velocity** have the same magnitude but point in opposite directions.
- Thus, the change in velocity (\(\Delta v\)) is:

\[
\Delta v = 2v
\]

where \(v\) is the magnitude of the velocity.

The magnitude of velocity (\(v\)) is the distance divided by the time:

\[
v = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2\pi}{0.05} = 40\pi \, \text{km/hr}
\]

Thus:

\[
\Delta v = 2(40\pi) = 80\pi \, \text{km/hr}
\]

#### Total Time:
The total time remains \(0.05 \, \text{hours}\).

#### Average Acceleration:
Substitute the values into the formula:

\[
\text{Average Acceleration} = \frac{\Delta v}{\Delta t} = \frac{80\pi}{0.05} = 1600\pi \, \text{km/hr}^2 \approx 5026.55 \, \text{km/hr}^2
\]

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### Final Results:
1. **Average Speed:** \(125.66 \, \text{km/hr}\)
2. **Average Acceleration:** \(5026.55 \, \text{km/hr}^2\)

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Do you have any questions or need further clarification? 😊

### Related Questions:
1. How would the average speed change if the radius of the semicircle were doubled?
2. What would be the average acceleration if the car traveled a full circular path?
3. How can we express the result in SI units (meters per second and meters per second squared)?
4. How does the curvature of the path affect the average acceleration of the car?
5. What role does centripetal acceleration play in this motion?

### Tip:
Always remember to distinguish between **scalar quantities** (e.g., speed) and **vector quantities** (e.g., velocity and acceleration) while solving motion problems.

Learn how to calculate the average speed and acceleration of a car traveling a semicircular path with a radius of 2 km in 3 minutes. This detailed solution explores circular motion concepts and includes step-by-step calculations using key kinematic formulas. Ideal for high school students in Grades 10-12.

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