A car moves from `X` to `Y` with a uniform speed `v_u` and returns to `Y` with a uniform speed `v_d`. The average speed for this round trip is :

Options

A. `(2 v_d v_u)/(v_d + v_u)`
B. `sqrt(v_u v_d)`
C. `(v_d v_u)/(v_d + v_u)`
D. `(v_u + v_d)/(2)`

Explanation

To find the average speed for the round trip of the car moving from point X to point Y and back, we can follow these steps:

### Step 1: Define the distances and speeds
Let the distance from X to Y be \( d \). The car travels from X to Y with a uniform speed \( v_u \) and returns from Y to X with a uniform speed \( v_d \).

### Step 2: Calculate the time taken for each leg of the trip
- The time taken to travel from X to Y (going) is given by:
\[ t_u = \frac{d}{v_u} \]
- The time taken to return from Y to X (returning) is given by:
\[ t_d = \frac{d}{v_d} \]

### Step 3: Calculate the total time for the round trip
The total time for the round trip is the sum of the time taken for each leg:
\[ t_{total} = t_u + t_d = \frac{d}{v_u} + \frac{d}{v_d} \]

### Step 4: Calculate the total distance for the round trip
The total distance covered in the round trip is:
\[ d_{total} = d + d = 2d \]

### Step 5: Calculate the average speed
The average speed \( v_{avg} \) is defined as the total distance divided by the total time:
\[ v_{avg} = \frac{d_{total}}{t_{total}} = \frac{2d}{\frac{d}{v_u} + \frac{d}{v_d}} \]

### Step 6: Simplify the expression for average speed
We can simplify the expression:
\[ v_{avg} = \frac{2d}{d\left(\frac{1}{v_u} + \frac{1}{v_d}\right)} = \frac{2d}{d \cdot \left(\frac{v_d + v_u}{v_u v_d}\right)} = \frac{2v_u v_d}{v_u + v_d} \]

### Final Result
Thus, the average speed for the round trip is:
\[ v_{avg} = \frac{2v_u v_d}{v_u + v_d} \]

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