If particle travels `n` equal distances with speeds `v_(1), v_(2),…v_(n)`, then the average speed `vec v` of the particle will be such that .

Options

A. `vecV=(v_(1) +v_(2) +.......+v_(n))/(n)`
B. `vecV=(nv_(1)v_(2)+v_(n))/(v_(1)+v_(2)+v_(3)+....+v_(n))`
C. `(1)/(barV)=(1)/(n) ((1)/(v_(1))+(1)/(v_(2))+.......+(1)/(v_(n)))`
D. `barV=sqrt(v_(1)^(2)+v_(2)^(2)+.....+(1)/(v_(n)))`

Explanation

To find the average speed \( \vec{v} \) of a particle that travels \( n \) equal distances with speeds \( v_1, v_2, \ldots, v_n \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
The total distance \( D \) traveled by the particle is given by:
\[ D = n \cdot s \]
where \( s \) is the distance of each segment.

2. **Calculate Time for Each Segment**:
The time taken to travel each segment can be calculated using the formula:
\[ t_i = \frac{s}{v_i} \]
for \( i = 1, 2, \ldots, n \). Thus, the total time \( T \) taken for all segments is:
\[ T = t_1 + t_2 + \ldots + t_n = \frac{s}{v_1} + \frac{s}{v_2} + \ldots + \frac{s}{v_n} \]

3. **Factor Out Common Terms**:
We can factor out \( s \) from the total time:
\[ T = s \left( \frac{1}{v_1} + \frac{1}{v_2} + \ldots + \frac{1}{v_n} \right) \]

4. **Calculate Average Speed**:
The average speed \( \vec{v} \) is defined as the total distance divided by the total time:
\[ \vec{v} = \frac{D}{T} = \frac{n \cdot s}{s \left( \frac{1}{v_1} + \frac{1}{v_2} + \ldots + \frac{1}{v_n} \right)} \]

5. **Simplify the Expression**:
Cancelling \( s \) from the numerator and denominator, we get:
\[ \vec{v} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \ldots + \frac{1}{v_n}} \]

6. **Final Result**:
Therefore, the average speed \( \vec{v} \) of the particle is given by:
\[ \vec{v} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \ldots + \frac{1}{v_n}} \]