Measure of two quantities along with the precision of respective measuring instrument is `A=2.5ms^(-1)+-0.5ms^(-1)`,B=0.10s+-0.01s`. The value of AB will be

Options

A. `(0.25+-0.08)m`
B. `(0.25+-0.5)m`
C. `(0.25+-0.05)m`
D. `(0.25+-0.135)m`

Explanation

To solve the problem, we need to find the product of two quantities \( A \) and \( B \), taking into account their uncertainties. Here’s a step-by-step solution:

### Step 1: Identify the given values
We have:
- \( A = 2.5 \, \text{ms}^{-1} \) with an uncertainty \( \Delta A = 0.5 \, \text{ms}^{-1} \)
- \( B = 0.10 \, \text{s} \) with an uncertainty \( \Delta B = 0.01 \, \text{s} \)

### Step 2: Calculate the product \( AB \)
To find the product \( X = AB \):
\[ X = A \times B = 2.5 \, \text{ms}^{-1} \times 0.10 \, \text{s} = 0.25 \, \text{m} \]

### Step 3: Calculate the relative uncertainties
The relative uncertainty in the product \( X \) can be calculated using the formula:
\[ \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]

Substituting the values:
\[ \frac{\Delta A}{A} = \frac{0.5}{2.5} = 0.2 \]
\[ \frac{\Delta B}{B} = \frac{0.01}{0.10} = 0.1 \]

Now, adding these relative uncertainties:
\[ \frac{\Delta X}{X} = 0.2 + 0.1 = 0.3 \]

### Step 4: Calculate the absolute uncertainty \( \Delta X \)
Now we can find the absolute uncertainty \( \Delta X \):
\[ \Delta X = X \times \frac{\Delta X}{X} = 0.25 \times 0.3 = 0.075 \]

### Step 5: Write the final result
The final result for \( AB \) with its uncertainty is:
\[ AB = 0.25 \pm 0.075 \, \text{m} \]

### Summary
Thus, the value of \( AB \) is \( 0.25 \, \text{m} \) with an uncertainty of \( 0.075 \, \text{m} \).

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