A motor vehicle travelled the first third of a distance s at a speed of `v_1`=10 kmph, the second third at a speed of `v_2`=20 kmph and the last third at a speed of `v_3`=60 kmph. Determine the mean speed of the vehicle over the entire distance s.

Options

A. 15 kmph
B. 12 kmph
C. 10 kmph
D. 18 kmph

Answer

To find the mean speed of the vehicle over the entire distance \( s \), we can follow these steps:

### Step 1: Define the total distance
Let the total distance \( s \) be divided into three equal parts. We can express \( s \) as:
\[ s = 3x \]
where \( x \) is the distance covered in each third of the total distance.

### Step 2: Calculate the time taken for each segment
The time taken to travel each segment can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

1. **First third** (distance = \( x \), speed = \( v_1 = 10 \) km/h):
\[ t_1 = \frac{x}{10} \]

2. **Second third** (distance = \( x \), speed = \( v_2 = 20 \) km/h):
\[ t_2 = \frac{x}{20} \]

3. **Last third** (distance = \( x \), speed = \( v_3 = 60 \) km/h):
\[ t_3 = \frac{x}{60} \]

### Step 3: Calculate the total time taken
Now, we can find the total time \( T \) taken for the entire journey by adding the times for each segment:
\[ T = t_1 + t_2 + t_3 = \frac{x}{10} + \frac{x}{20} + \frac{x}{60} \]

To add these fractions, we need a common denominator. The least common multiple of 10, 20, and 60 is 60. Rewriting each term:
\[ T = \frac{6x}{60} + \frac{3x}{60} + \frac{x}{60} = \frac{(6 + 3 + 1)x}{60} = \frac{10x}{60} = \frac{x}{6} \]

### Step 4: Calculate the mean speed
The mean speed \( V_{mean} \) is defined as the total distance divided by the total time:
\[ V_{mean} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{s}{T} \]
Substituting the values we have:
\[ V_{mean} = \frac{3x}{\frac{x}{6}} = 3x \cdot \frac{6}{x} = 18 \text{ km/h} \]

### Final Answer
The mean speed of the vehicle over the entire distance \( s \) is:
\[ \boxed{18 \text{ km/h}} \]

---