Solving Constable and Thief Relative Speed Problem

This problem involves understanding the concept of relative speed. When two objects are moving in the same direction, the relative speed at which the faster object gains on the slower object is the difference between their speeds.

Understanding the Problem Setup

We have a constable, Ram, chasing a thief, Somesh. Ram is initially behind Somesh. We are given their speeds and the initial distance between them. We need to find the time it takes for Ram to catch Somesh.

• Initial distance between Ram and Somesh = 225 m
• Ram's speed = 50 m/minute
• Somesh's speed = 30 m/minute

Calculating the Relative Speed

Since Ram and Somesh are moving in the same direction, Ram closes the distance between them at a rate equal to the difference in their speeds. This is the relative speed.

Relative Speed = Speed of the faster object - Speed of the slower object

In this case:

Relative Speed = Ram's speed - Somesh's speed

Relative Speed $= (50 - 30)$ m/minute

Relative Speed $= 20$ m/minute

This means Ram reduces the distance between himself and Somesh by 20 meters every minute.

Calculating the Time to Catch Up

Ram needs to cover the initial distance of 225 meters at the relative speed of 20 m/minute. The time taken can be calculated using the formula:

Time $= \frac{\text{Distance}}{\text{Speed}}$

Here, the distance is the initial gap, and the speed is the relative speed.

Time $= \frac{225 \text{ m}}{20 \text{ m/minute}}$

Time $= 11.25$ minutes

Converting Time to Minutes and Seconds

The time calculated is 11.25 minutes. This means 11 full minutes and 0.25 of a minute.

To convert the decimal part (0.25 minutes) into seconds, we multiply by 60 (since there are 60 seconds in a minute).

Seconds $= 0.25 \times 60$ seconds

Seconds $= 15$ seconds

So, the total time taken for Constable Ram to catch thief Somesh is 11 minutes and 15 seconds.

Conclusion

Constable Ram will catch thief Somesh after 11 minutes and 15 seconds.

Revision Table: Key Concepts

Review the main ideas used in solving this relative speed problem.

• Relative Speed: When objects move in the same direction, relative speed is the difference in their speeds ($v_{rel} = v_1 - v_2$). When objects move in opposite directions, relative speed is the sum of their speeds ($v_{rel} = v_1 + v_2$).
• Distance, Speed, Time Relation: The fundamental formula is $\text{Distance} = \text{Speed} \times \text{Time}$. This can be rearranged to find speed ($\text{Speed} = \frac{\text{Distance}}{\text{Time}}$) or time ($\text{Time} = \frac{\text{Distance}}{\text{Speed}}$).
• Catching Up Problems: These are typically solved using the concept of relative speed in the same direction. The distance to be covered is the initial gap between the objects.

Additional Information on Relative Speed Problems

Relative speed is a crucial concept in time and distance problems. It simplifies scenarios where two or more objects are in motion.

• Same Direction: If two objects A and B move in the same direction with speeds $v_A$ and $v_B$ respectively ($v_A > v_B$), their relative speed is $v_A - v_B$. This is the rate at which A gains on B.
• Opposite Directions: If two objects A and B move towards each other with speeds $v_A$ and $v_B$, their relative speed is $v_A + v_B$. This is the rate at which the distance between them decreases. If they move away from each other, the rate at which the distance increases is also $v_A + v_B$.
• Meeting Point: In problems involving meeting points, the total distance covered by both objects combined is the initial distance between them (when moving towards each other). The time taken to meet is $\frac{\text{Initial Distance}}{\text{Relative Speed (towards each other)}}$.
• Trains: Problems involving trains often require considering the length of the trains. When a train crosses a pole or a person, the distance covered is the length of the train. When a train crosses a platform or another train, the distance covered is the sum of the length of the train and the length of the platform or the other train.

Understanding relative speed makes solving complex motion problems much easier by reducing the motion of two objects into the motion of one object relative to the other.