Policeman Thief Chase Problem Explained
This problem involves a classic scenario of relative motion where one object (the policeman) is chasing another (the thief). We need to determine how far the thief runs before the policeman catches up.
Chase Information Details
Let's list the details provided in the question:
| Parameter | Value |
|---|---|
| Initial Distance (D) | 100 metres |
| Thief's Speed ($v_t$) | 8 km/h |
| Policeman's Speed ($v_p$) | 10 km/h |
Distance Thief Runs Before Overtake
To solve this, we need to find out how long it takes for the policeman to close the initial gap and then calculate the distance the thief covers in that time.
Speed Conversion for Chase
Since the distance is given in metres, it's easier to work with speeds in m/s. We know that 1 km/h = $\frac{5}{18}$ m/s.
- Policeman's speed: $v_p = 10 \text{ km/h} = 10 \times \frac{5}{18} \text{ m/s} = \frac{50}{18} \text{ m/s} = \frac{25}{9} \text{ m/s}$
- Thief's speed: $v_t = 8 \text{ km/h} = 8 \times \frac{5}{18} \text{ m/s} = \frac{40}{18} \text{ m/s} = \frac{20}{9} \text{ m/s}$
Relative Speed Calculation
The policeman is faster than the thief. The speed at which the policeman gains on the thief is the difference between their speeds. This is called the relative speed ($v_{rel}$).
Relative Speed Calculation:
$$v_{rel} = v_p - v_t$$ $$v_{rel} = \frac{25}{9} \text{ m/s} - \frac{20}{9} \text{ m/s}$$ $$v_{rel} = \frac{5}{9} \text{ m/s}$$This means the distance between the policeman and the thief decreases by $\frac{5}{9}$ metres every second.
Time to Overtake Calculation
The policeman needs to cover the initial distance of 100 metres at this relative speed. We can use the formula: Time = Distance / Speed.
Time Calculation:
$$t = \frac{\text{Initial Distance}}{v_{rel}}$$ $$t = \frac{100 \text{ m}}{\frac{5}{9} \text{ m/s}}$$ $$t = 100 \times \frac{9}{5} \text{ s}$$ $$t = 20 \times 9 \text{ s}$$ $$t = 180 \text{ seconds}$$So, it will take 180 seconds for the policeman to catch the thief.
Thief Distance Calculation
Now we need to find out how far the thief ran during these 180 seconds. We use the formula: Distance = Speed × Time.
Thief's Distance Calculation:
$$d_t = v_t \times t$$ $$d_t = \frac{20}{9} \text{ m/s} \times 180 \text{ s}$$ $$d_t = 20 \times \frac{180}{9} \text{ m}$$ $$d_t = 20 \times 20 \text{ m}$$ $$d_t = 400 \text{ metres}$$Thief Overtake Distance Result
The thief will have run 400 metres before the policeman overtakes him.
Similar Questions
- A policeman follows a thief who is 400 m ahead of him. If they run at the speed of 7 km/h and 6 km/h, respectively, what distance does the policeman run to catch up with the thief?
- A thief is noticed by a policeman from a distance of 800 m. The thief starts running and the policeman chases him. The thief and the policeman run at speeds of 20 km/h and 25 km/h, respectively. What is the distance between them after 9 minutes?
A policeman chases a thief on a straight road. The policeman catches the thief in 15 minutes if he drives with a speed of 60 km/h and catches the thief in 0.5 hours if he drives with a speed 45 km/h. What should be his driving speed (in km/h) to catch the thief in just 10 minutes?
Important Questions from Relative Speed
- Ankur jogs on a circular track X at $1 \frac{2}{3}$ m/s and Mahima jogs on track Y at $2 \frac{1}{12}$ m/s. The total length of Tracks X and Y is 0.325 km. Ankur makes 9 rounds of track X in the same time Mahima makes 5 rounds of track Y. The time taken (in seconds) by Mahima to complete one round of track X is:
- Two places, A and B are 3710 kms from each other. A train leaves from A to B at the same time another train leaves B to A. The two trains meet at the end of 35 hours. If the train travelling from A to B travels 35 kmph faster than the other. Find the speed of the faster train. (In kmph)
- A truck, 1575 metres long is running at a speed of 109 km/hr. How many seconds will it take to cross a 1475 metres long train running in the same direction at a speed of 79 km/hr?
- A truck, 1600 metres long is running at a speed of 112 km/hr. How many seconds will it take to cross a 1500 metres long train running in the same direction at a speed of 82 km/hr?
- Two friends 10 km apart start running towards each other at speeds of 10 km/hr and 14 km/hr respectively. After how much time will they meet each other?
