A body travels uniformly a distance of `( 13.8 +- 0.2) m` in a time `(4.0 +- 0.3) s`. Find the velocity of the body within error limits and the percentage error.

Explanation

To find the velocity of the body along with the error limits and the percentage error, we can follow these steps:

### Step 1: Calculate the Velocity
The formula for velocity \( V \) is given by:

\[ V = \frac{S}{T} \]

where \( S \) is the distance traveled and \( T \) is the time taken.

Given:
- Distance \( S = 13.8 \, m \)
- Time \( T = 4.0 \, s \)

Substituting the values:

\[ V = \frac{13.8 \, m}{4.0 \, s} = 3.45 \, m/s \]

### Step 2: Calculate the Absolute Errors
The absolute errors in distance and time are given as:
- \( \Delta S = 0.2 \, m \)
- \( \Delta T = 0.3 \, s \)

### Step 3: Calculate the Relative Errors
The relative error in velocity can be calculated using the formula:

\[ \frac{\Delta V}{V} = \frac{\Delta S}{S} + \frac{\Delta T}{T} \]

Substituting the known values:

\[ \frac{\Delta V}{3.45} = \frac{0.2}{13.8} + \frac{0.3}{4.0} \]

Calculating each term:

1. \( \frac{0.2}{13.8} \approx 0.0145 \)
2. \( \frac{0.3}{4.0} = 0.075 \)

Adding these together:

\[ \frac{\Delta V}{3.45} \approx 0.0145 + 0.075 = 0.0895 \]

### Step 4: Calculate the Absolute Error in Velocity
Now, we can find \( \Delta V \):

\[ \Delta V = 3.45 \times 0.0895 \approx 0.308 \, m/s \]

Rounding this to one decimal place gives us:

\[ \Delta V \approx 0.3 \, m/s \]

### Step 5: Final Result
The velocity of the body with the error limits is:

\[ V = 3.45 \pm 0.3 \, m/s \]

### Step 6: Calculate the Percentage Error
The percentage error can be calculated using the formula:

\[ \text{Percentage Error} = \left( \frac{\Delta V}{V} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage Error} = \left( \frac{0.3}{3.45} \right) \times 100 \approx 8.70\% \]

### Final Answer
Thus, the velocity of the body is:

\[ 3.45 \pm 0.3 \, m/s \quad \text{with a percentage error of } 8.70\% \]

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