Rachel walks on a straight road from her home to a gymnasium 2.80 km away with a speed of 6.00 km/h. As soon as she reaches the gymnasium, she immediately turns and walks back home with a speed of 7.70 km/h as she finds the gymnasium closed. What are the (a) magnitude of average velocity and (b) average speed of Rechel over the interval of time 0.00-35.0 min ?

Answer

To solve the problem, we need to calculate both the average velocity and average speed for Rachel's journey from her home to the gymnasium and back. Let's break it down step by step.

### Step 1: Calculate the time taken to walk to the gymnasium
Rachel walks a distance of 2.80 km to the gymnasium at a speed of 6.00 km/h.

\[ \text{Time to gym} (T_1) = \frac{\text{Distance}}{\text{Speed}} = \frac{2.80 \text{ km}}{6.00 \text{ km/h}} = 0.4667 \text{ hours} \approx 28 \text{ minutes} \]

### Step 2: Calculate the time taken to walk back home
Rachel walks back home at a speed of 7.70 km/h.

\[ \text{Time to return} (T_2) = \frac{\text{Distance}}{\text{Speed}} = \frac{2.80 \text{ km}}{7.70 \text{ km/h}} = 0.3636 \text{ hours} \approx 21.82 \text{ minutes} \]

### Step 3: Calculate the total time of the journey
The total time taken for the entire journey is the sum of the time taken to go to the gym and the time taken to return home.

\[ \text{Total time} (T) = T_1 + T_2 = 0.4667 \text{ hours} + 0.3636 \text{ hours} = 0.8303 \text{ hours} \approx 49.82 \text{ minutes} \]

### Step 4: Calculate the total displacement
Displacement is defined as the change in position. Since Rachel returns to her starting point (home), the total displacement is 0 km.

\[ \text{Total displacement} = \text{Final position} - \text{Initial position} = 0 \text{ km} \]

### Step 5: Calculate the average velocity
Average velocity is defined as the total displacement divided by the total time taken.

\[ \text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} = \frac{0 \text{ km}}{0.8303 \text{ hours}} = 0 \text{ km/h} \]

### Step 6: Calculate the total distance traveled
Total distance is the sum of the distance to the gym and the distance back home.

\[ \text{Total distance} = 2.80 \text{ km} + 2.80 \text{ km} = 5.60 \text{ km} \]

### Step 7: Calculate the average speed
Average speed is defined as the total distance traveled divided by the total time taken.

\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{5.60 \text{ km}}{0.8303 \text{ hours}} \approx 6.75 \text{ km/h} \]

### Final Answers
(a) The magnitude of average velocity is **0 km/h**.
(b) The average speed is approximately **6.75 km/h**.

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