A body travels uniformly a distance of `(13.8pm0.2)` m in a time `(4.0pm0.3)` S. The velocity of the body within error limit is

Options

A. `(3.45pm0.2)" ms"^(-1)`
B. `(3.45pm0.3)" ms"^(-1)`
C. `(3.45pm0.4)" ms"^(-1)`
D. `(3.45pm0.5)" ms"^(-1)`

Answer

To find the velocity of the body within error limits, we will follow these steps:

### Step 1: Calculate the main value of 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 \( s = 13.8 \, \text{m} \)
- Time \( t = 4.0 \, \text{s} \)

Substituting the values:

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

### Step 2: Calculate the relative errors
To calculate the error in velocity, we need to find the relative errors in distance and time.

Given:
- Error in distance \( \Delta s = 0.2 \, \text{m} \)
- Error in time \( \Delta t = 0.3 \, \text{s} \)

The relative error in distance \( \frac{\Delta s}{s} \) and time \( \frac{\Delta t}{t} \) can be calculated as follows:

\[ \frac{\Delta s}{s} = \frac{0.2}{13.8} \]
\[ \frac{\Delta t}{t} = \frac{0.3}{4.0} \]

Calculating these values:

\[ \frac{\Delta s}{s} = \frac{0.2}{13.8} \approx 0.01449 \]
\[ \frac{\Delta t}{t} = \frac{0.3}{4.0} = 0.075 \]

### Step 3: Calculate the total relative error in velocity
The total relative error in velocity \( \frac{\Delta v}{v} \) is given by:

\[ \frac{\Delta v}{v} = \frac{\Delta s}{s} + \frac{\Delta t}{t} \]

Substituting the values:

\[ \frac{\Delta v}{v} = 0.01449 + 0.075 \approx 0.08949 \]

### Step 4: Calculate the absolute error in velocity
Now, we can find the absolute error \( \Delta v \):

\[ \Delta v = v \times \frac{\Delta v}{v} \]

Substituting the values:

\[ \Delta v = 3.45 \times 0.08949 \approx 0.308 \]

### Step 5: Present the final result
The velocity of the body within error limits can be expressed as:

\[ v = 3.45 \pm 0.31 \, \text{m/s} \]

### Summary
The final answer is:

\[ \text{Velocity} = 3.45 \pm 0.31 \, \text{m/s} \]