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) form `A` to `B` and (b) form `A` to `C`?

Explanation

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

### Step 1: Calculate Average Speed from A to B

1. **Distance from A to B**: The distance is given as 300 meters.
2. **Time taken from A to B**: The time is given as 2 minutes and 30 seconds. To convert this into seconds:
\[ 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]
\[ 30 \text{ seconds} = 30 \text{ seconds} \]
Therefore, total time:
\[ T_1 = 120 + 30 = 150 \text{ seconds} \]

3. **Average Speed Calculation**:
The formula for average speed is:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values:
\[ \text{Average Speed} = \frac{300 \text{ m}}{150 \text{ s}} = 2 \text{ m/s} \]

### Step 2: Calculate Average Velocity from A to B

1. **Displacement from A to B**: Since Joseph jogs in a straight line from A to B, the displacement is equal to the distance:
\[ \text{Displacement} = 300 \text{ m} \]

2. **Average Velocity Calculation**:
The formula for average velocity is:
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]
Substituting the values:
\[ \text{Average Velocity} = \frac{300 \text{ m}}{150 \text{ s}} = 2 \text{ m/s} \]

### Step 3: Calculate Average Speed from A to C

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

2. **Total Time from A to C**: The time taken from A to B is 150 seconds, and the time taken from B to C is 1 minute (60 seconds):
\[ T_2 = 60 \text{ seconds} \]
Therefore, total time:
\[ \text{Total Time} = 150 \text{ s} + 60 \text{ s} = 210 \text{ s} \]

3. **Average Speed Calculation**:
\[ \text{Average Speed} = \frac{400 \text{ m}}{210 \text{ s}} \approx 1.90 \text{ m/s} \]

### Step 4: Calculate Average Velocity from A to C

1. **Displacement from A to C**: The displacement is the straight line distance from the starting point A to the final point C. Since C is 200 m away from A (300 m - 100 m), the displacement is:
\[ \text{Displacement} = 200 \text{ m} \]

2. **Average Velocity Calculation**:
\[ \text{Average Velocity} = \frac{200 \text{ m}}{210 \text{ s}} \approx 0.95 \text{ m/s} \]

### Final Answers:

- Average Speed from A to B: **2 m/s**
- Average Velocity from A to B: **2 m/s**
- Average Speed from A to C: **1.90 m/s**
- Average Velocity from A to C: **0.95 m/s**