Step 1: Identify given values

The speed of the thief is given as $v_t = 8 \text{ km/h}$ and the speed of the policeman is $v_p = 10 \text{ km/h}$. The initial separation between them is $d = 100 \text{ m}$.

$v_t = 8 \text{ km/h}, \quad v_p = 10 \text{ km/h}, \quad d = 100 \text{ m}$

Step 2: Calculate the relative speed

Since both are moving in the same direction, the relative speed of the policeman with respect to the thief is the difference between their speeds: $v_{rel} = 10^{-8} = 2 \text{ km/h}$.

$v_{rel} = 10^{-8} = 2 \text{ km/h}$

Step 3: Convert relative speed to meters per minute

To match the distance in meters ($100 \text{ m}$), convert $2 \text{ km/h}$ to meters per minute: $2 \text{ km/h} = 2 \times \frac{1000 \text{ m}}{60 \text{ min}} = \frac{100}{3} \text{ m/min}$.

$v_{rel} = 2 \times \frac{1000}{60} = \frac{100}{3} \text{ m/min}$

Step 4: Calculate the time required

Using the formula $t = \frac{\text{Distance}}{\text{Relative Speed}}$, substitute the values to find the time: $t = \frac{100}{\frac{100}{3}} = 100 \times \frac{3}{100} = 3 \text{ minutes}$.

$t = \frac{100}{\frac{100}{3}} = 3 \text{ minutes}$