A student goes from his to his friend`s house with speed ` v_1`. Finding the door of his friend`s home closed, he returns back to his own house with the speed ` v_2`. Then the averge speed and net displacement of student is [consider distance between two houses be S].

Options

A. (a) ` (v_1+v_2)/2 , 0`
B. (b) ` sqrt ( v_1 v_2 , 2 S`
C. (c ) (2 v_1 v_2 )/(v_1+v_2), 0`
D. (d) ` v_1 v_2, S`

Explanation

To solve the problem step by step, we will analyze the student's journey from his home to his friend's house and back, calculating the average speed and net displacement.

### Step 1: Understand the Journey
The student travels from his home to his friend's house and then returns back. The distance between the two houses is given as \( S \).

### Step 2: Calculate Displacement
Displacement is defined as the change in position. Since the student starts and ends at the same point (his home), the net displacement is:
\[ \text{Net Displacement} = 0 \]

### Step 3: Calculate Time Taken for Each Leg of the Journey
1. **Time taken to go to friend's house**:
- Speed = \( v_1 \)
- Distance = \( S \)
- Time taken (T1) is given by:
\[ T_1 = \frac{S}{v_1} \]

2. **Time taken to return home**:
- Speed = \( v_2 \)
- Distance = \( S \)
- Time taken (T2) is given by:
\[ T_2 = \frac{S}{v_2} \]

### Step 4: Calculate Total Distance and Total Time
- **Total Distance**:
- The student travels to his friend's house and back, so the total distance is:
\[ \text{Total Distance} = S + S = 2S \]

- **Total Time**:
- The total time taken for the journey is the sum of the time taken for each leg:
\[ \text{Total Time} = T_1 + T_2 = \frac{S}{v_1} + \frac{S}{v_2} \]

### Step 5: Calculate Average Speed
Average speed is defined as the total distance divided by the total time taken:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2S}{\frac{S}{v_1} + \frac{S}{v_2}} \]

We can simplify this expression:
\[ \text{Average Speed} = \frac{2S}{S \left(\frac{1}{v_1} + \frac{1}{v_2}\right)} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} = \frac{2v_1v_2}{v_1 + v_2} \]

### Final Answers
- **Net Displacement**: \( 0 \)
- **Average Speed**: \( \frac{2v_1v_2}{v_1 + v_2} \)

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