Measure of two quantities along with the precision of respective measuring instrument is A = 2.5 m s–1 ± 0.5 m s–1 and B = 0.10 s ± 0.01 s. The value of A B will be

The correct option is (0.25 ± 0.08) m.

Let us understand how to calculate the product of two measured quantities along with its uncertainty (error limits) step-by-step.

Step 1: Identify the given values and their absolute uncertainties
The measured value of quantity A is:
A = 2.5 m s^-1
The absolute uncertainty in A is:
ΔA = 0.5 m s^-1

The measured value of quantity B is:
B = 0.10 s
The absolute uncertainty in B is:
ΔB = 0.01 s

Step 2: Calculate the nominal value of the product (Z = A * B)
Let Z = A * B.
Substituting the nominal values:
Z = 2.5 m s^-1 * 0.10 s = 0.25 m

Step 3: Determine the relative uncertainty in the product Z
When two quantities are multiplied, their fractional or relative errors add up. The relation is given by:

$$\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$

Let us substitute the values into this formula:

$$\frac{\Delta Z}{0.25}=\frac{0.5}{2.5}+\frac{0.01}{0.10}$$

Now, simplify the terms on the right-hand side:
First term:
0.5 / 2.5 = 1 / 5 = 0.20
Second term:
0.01 / 0.10 = 1 / 10 = 0.10
Adding them together:
0.20 + 0.10 = 0.30

Step 4: Calculate the absolute uncertainty (ΔZ)
Using the relative uncertainty:

$$\Delta Z=0.30*Z$$

$$\Delta Z=0.30*0.25=0.075\text{m}$$

Rounding the uncertainty to a standard format, we round 0.075 to 0.08 m (or looking at the given options, it matches 0.08 m).

Thus, expressing the value of the product A B with its uncertainty, we get:
A B = (0.25 ± 0.08) m