A particle is moving with uniform acceleration along a straight line ABC, where AB= BC. The average velocity of the particle from A to B is 10 m/s and from B to C is 15 m/s. The average velocity for the whole journey from A to C in m/s is

Options

A. 12
B. `12.5`
C. 13
D. `13.5`

Explanation

To find the average velocity of the particle moving from point A to point C, we will follow these steps:

### Step 1: Understand the Problem
We have a particle moving with uniform acceleration along a straight line from point A to point C, with point B in between. The distances AB and BC are equal. The average velocity from A to B (V_AB) is given as 10 m/s, and from B to C (V_BC) is given as 15 m/s.

### Step 2: Define Distances
Let the distance AB = BC = d. Therefore, the total distance from A to C is:
\[ d_{AC} = AB + BC = d + d = 2d \]

### Step 3: Calculate Time Taken for Each Segment
Using the formula for average velocity:
\[ \text{Average Velocity} = \frac{\text{Distance}}{\text{Time}} \]
We can rearrange this to find time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Average Velocity}} \]

#### Time from A to B (T1):
\[ T_1 = \frac{d}{V_{AB}} = \frac{d}{10} \]

#### Time from B to C (T2):
\[ T_2 = \frac{d}{V_{BC}} = \frac{d}{15} \]

### Step 4: Calculate Total Time
The total time (T) taken for the journey from A to C is:
\[ T = T_1 + T_2 = \frac{d}{10} + \frac{d}{15} \]

To add these fractions, we need a common denominator. The least common multiple of 10 and 15 is 30.
\[ T = \frac{3d}{30} + \frac{2d}{30} = \frac{5d}{30} = \frac{d}{6} \]

### Step 5: Calculate Average Velocity from A to C
Now we can find the average velocity (V_AC) for the entire journey from A to C:
\[ V_{AC} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{T} \]
Substituting T:
\[ V_{AC} = \frac{2d}{\frac{d}{6}} = 2d \times \frac{6}{d} = 12 \text{ m/s} \]

### Final Answer
The average velocity for the whole journey from A to C is:
\[ \boxed{12 \text{ m/s}} \]

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