Joseph jogs from one end `A` to other end `B` of a straight `300m` road in `2` minutes `30` seconds and then turns around and jogs `100 m` back to point `C` in another 1 minute. What are Joseph's average speeds and velocities in Jogging (a) from `A` to `B` and (b) from `A` to `C`?

Explanation

To solve the problem, we will calculate Joseph's average speed and average velocity for both segments of his jog: from point A to point B and from point A to point C.

### Part (a): From A to B

1. **Calculate the Distance from A to B**:
- The distance from point A to point B is given as **300 meters**.

2. **Calculate the Time Taken from A to B**:
- The time taken is **2 minutes 30 seconds**.
- Convert this time into seconds:
\[ 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]
\[ 30 \text{ seconds} = 30 \text{ seconds} \]
\[ \text{Total time} = 120 + 30 = 150 \text{ seconds} \]

3. **Calculate Average Speed from A to B**:
- Average speed is defined as total distance divided by total time.
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{300 \text{ meters}}{150 \text{ seconds}} = 2 \text{ m/s} \]

4. **Calculate Average Velocity from A to B**:
- Average velocity is defined as total displacement divided by total time.
- Since Joseph starts at A and ends at B, the displacement is equal to the distance:
\[ \text{Displacement} = 300 \text{ meters} \]
\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}} = \frac{300 \text{ meters}}{150 \text{ seconds}} = 2 \text{ m/s} \]

### Part (b): From A to C

1. **Calculate the Total Distance from A to C**:
- Joseph jogs from A to B (300 meters) and then back to C (100 meters).
- The total distance from A to C is:
\[ \text{Total Distance} = 300 + 100 = 400 \text{ meters} \]

2. **Calculate the Total Time Taken from A to C**:
- Time from A to B is 150 seconds (2 minutes 30 seconds).
- Time from B to C is **1 minute**, which is:
\[ 1 \text{ minute} = 60 \text{ seconds} \]
- Total time taken:
\[ \text{Total Time} = 150 + 60 = 210 \text{ seconds} \]

3. **Calculate Average Speed from A to C**:
- Average speed is total distance divided by total time.
\[ \text{Average Speed} = \frac{400 \text{ meters}}{210 \text{ seconds}} \approx 1.90 \text{ m/s} \]

4. **Calculate Average Velocity from A to C**:
- The displacement from A to C is the straight-line distance from A to C.
- Since Joseph jogged 300 meters to B and then 100 meters back to C, the displacement is:
\[ \text{Displacement} = 300 - 100 = 200 \text{ meters} \]
- Average velocity is displacement divided by total time:
\[ \text{Average Velocity} = \frac{200 \text{ meters}}{210 \text{ seconds}} \approx 0.95 \text{ m/s} \]

### Summary of Results:
- From A to B:
- Average Speed: **2 m/s**
- Average Velocity: **2 m/s**

- From A to C:
- Average Speed: **1.90 m/s**
- Average Velocity: **0.95 m/s**