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.

Options

A. `(3.5+-0.6)ms^(-1)`
B. `(3.5+-0.3)ms^(-1)`
C. `(6.1+-0.6)ms^(-1)`
D. `(6.1+-0.3)ms^(-1)`

Explanation

To solve the problem of finding the velocity of a body traveling a distance with associated uncertainties, we can follow these steps:

### Step 1: Identify the given values
- Distance \( S = 13.8 \, \text{m} \) with uncertainty \( \Delta S = 0.2 \, \text{m} \)
- Time \( T = 4.0 \, \text{s} \) with uncertainty \( \Delta T = 0.3 \, \text{s} \)

### Step 2: Calculate the velocity
The formula for velocity \( V \) is given by:
\[ V = \frac{S}{T} \]
Substituting the values:
\[ V = \frac{13.8 \, \text{m}}{4.0 \, \text{s}} = 3.45 \, \text{m/s} \]
Rounding to two significant figures, we get:
\[ V \approx 3.5 \, \text{m/s} \]

### Step 3: Calculate the relative uncertainties
The relative uncertainty in velocity \( \frac{\Delta V}{V} \) can be calculated using the formula:
\[ \frac{\Delta V}{V} = \frac{\Delta S}{S} + \frac{\Delta T}{T} \]
Substituting the values:
\[ \frac{\Delta V}{V} = \frac{0.2 \, \text{m}}{13.8 \, \text{m}} + \frac{0.3 \, \text{s}}{4.0 \, \text{s}} \]

Calculating each term:
1. \( \frac{0.2}{13.8} \approx 0.01449 \)
2. \( \frac{0.3}{4.0} = 0.075 \)

Adding these:
\[ \frac{\Delta V}{V} \approx 0.01449 + 0.075 = 0.08949 \]

### Step 4: Calculate the absolute uncertainty in velocity
To find \( \Delta V \):
\[ \Delta V = V \times \frac{\Delta V}{V} \]
Substituting the values:
\[ \Delta V = 3.5 \, \text{m/s} \times 0.08949 \approx 0.3122 \, \text{m/s} \]
Rounding to one decimal place, we get:
\[ \Delta V \approx 0.3 \, \text{m/s} \]

### Step 5: Write the final result
The velocity of the body within error limits is:
\[ V = 3.5 \pm 0.3 \, \text{m/s} \]

### Step 6: Calculate the percentage error
The percentage error can be calculated using:
\[ \text{Percentage Error} = \left( \frac{\Delta V}{V} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Error} = 0.08949 \times 100 \approx 8.95\% \]

### Final Answer
The velocity of the body is \( 3.5 \pm 0.3 \, \text{m/s} \) with a percentage error of approximately \( 8.95\% \).

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