A body moving along a straight line travels one third of the total distance with a speed of `3.0 m s^(-1)`. The remaining distance is covered with a speed of `4.0 m s^(-1)` for half the time and `5.0 m s^(-1)` for the other half of the time. The average speed during the motion is

Options

A. `4.0 m s^(-1)`
B. `6.0 m s^(-1)`
C. `3.8 m s^(-1)`
D. `2.4 m s^(-1)`

Explanation

To solve the problem step by step, we will break down the information provided and calculate the average speed of the body during its motion.

### Step 1: Define the total distance
Let the total distance traveled by the body be \( S \).

### Step 2: Calculate the distance for the first part
The body travels one third of the total distance at a speed of \( 3.0 \, \text{m/s} \). Therefore, the distance covered in the first part is:
\[ d_1 = \frac{S}{3} \]

### Step 3: Calculate the time taken for the first part
The time taken to cover this distance can be calculated using the formula:
\[ t_1 = \frac{d_1}{\text{speed}} = \frac{\frac{S}{3}}{3} = \frac{S}{9} \]

### Step 4: Calculate the remaining distance
The remaining distance is:
\[ d_2 = S - d_1 = S - \frac{S}{3} = \frac{2S}{3} \]

### Step 5: Determine the time distribution for the remaining distance
The remaining distance \( d_2 \) is covered in two halves of time:
- Half the time at \( 4.0 \, \text{m/s} \)
- The other half at \( 5.0 \, \text{m/s} \)

Let the total time for the remaining distance be \( T \). Therefore, each half time is \( \frac{T}{2} \).

### Step 6: Calculate the distances covered in each half
The distance covered in the first half (at \( 4.0 \, \text{m/s} \)):
\[ d_{2,1} = \text{speed} \times \text{time} = 4 \times \frac{T}{2} = 2T \]

The distance covered in the second half (at \( 5.0 \, \text{m/s} \)):
\[ d_{2,2} = \text{speed} \times \text{time} = 5 \times \frac{T}{2} = \frac{5T}{2} \]

### Step 7: Set up the equation for the remaining distance
The total distance for the remaining part is:
\[ d_{2,1} + d_{2,2} = 2T + \frac{5T}{2} = \frac{4T}{2} + \frac{5T}{2} = \frac{9T}{2} \]
Setting this equal to the remaining distance:
\[ \frac{9T}{2} = \frac{2S}{3} \]

### Step 8: Solve for \( T \)
To solve for \( T \), multiply both sides by 2:
\[ 9T = \frac{4S}{3} \]
Now divide both sides by 9:
\[ T = \frac{4S}{27} \]

### Step 9: Calculate the total time
The total time \( T_{total} \) is the sum of the time for both parts:
\[ T_{total} = t_1 + T = \frac{S}{9} + \frac{4S}{27} \]
To add these fractions, find a common denominator (which is 27):
\[ T_{total} = \frac{3S}{27} + \frac{4S}{27} = \frac{7S}{27} \]

### Step 10: Calculate the average speed
The average speed \( V_{avg} \) is given by the total distance divided by the total time:
\[ V_{avg} = \frac{S}{T_{total}} = \frac{S}{\frac{7S}{27}} = \frac{27}{7} \, \text{m/s} \]

### Step 11: Final Calculation
Calculating \( \frac{27}{7} \):
\[ \frac{27}{7} \approx 3.857 \, \text{m/s} \]

### Conclusion
Thus, the average speed during the motion is approximately \( 3.86 \, \text{m/s} \).