Policeman Chase Distance Calculation

This problem requires us to calculate the total distance covered by a policeman until they catch a thief, based on their initial separation and running speeds.

Problem Setup Analysis

We are given the following information:

• Initial separation distance between policeman and thief: 400 meters.
• Policeman's speed: 7 km/h.
• Thief's speed: 6 km/h.
• Both are moving in the same direction.

The goal is to find the distance the policeman runs before catching the thief.

Understanding Relative Speed in Chases

When two objects move in the same direction, the speed at which the faster object gains on the slower object is called the relative speed. It's calculated by subtracting the speed of the slower object from the speed of the faster object.

Let $v_p$ represent the speed of the policeman and $v_t$ represent the speed of the thief.

Given:

• $v_p = 7$ km/h
• $v_t = 6$ km/h

The relative speed ($v_{rel}$) of the policeman with respect to the thief is:

$v_{rel} = v_p - v_t$

$v_{rel} = 7 \text{ km/h} - 6 \text{ km/h} = 1 \text{ km/h}$

This means the policeman closes the distance gap by 1 kilometer every hour.

Calculating the Time to Catch Up

The time it takes for the policeman to catch the thief is found by dividing the initial distance between them by their relative speed.

Time ($t$) = $\frac{\text{Initial Distance}}{\text{Relative Speed}}$

Before calculating, we need consistent units. Let's convert the initial distance from meters to kilometers:

Initial Distance = 400 m = $\frac{400}{1000}$ km = $0.4$ km.

Now, we calculate the time:

$t = \frac{0.4 \text{ km}}{1 \text{ km/h}} = 0.4$ hours.

Therefore, the policeman needs 0.4 hours to catch the thief.

Determining the Policeman's Total Running Distance

To find the distance the policeman runs, we multiply the policeman's speed by the time taken to catch up.

Distance run by Policeman = Policeman's Speed $\times$ Time taken

Distance = $v_p \times t$

Distance = $7 \text{ km/h} \times 0.4 \text{ hours}$

Distance = $2.8$ km.

The options are given in meters, so we convert the distance back to meters:

Distance = $2.8 \text{ km} \times 1000 \text{ m/km} = 2800 \text{ m}$.

Final Answer Summary

The policeman runs a distance of 2800 meters to catch up with the thief.