Question

There are fuo places \(\mathrm{X}\) and \(Y, 200 \mathrm{~km}\) apart from each other. Initially two persons \(P\) and \(Q\) both are at \(^{\circ} X:\) The speed of \(P\) is \(20 \mathrm{~km} / \mathrm{h}\) and speed of \(Q\) is \(30 \mathrm{~km} / \mathrm{h}\). Later on they starts to move to and fro benveen \(X\) and \(Y\). If they meet second time each other at a point \(N\) somewhere between \(X\) and \(Y\), then the distance travelled by \(Q\) is : (a) \(240 \mathrm{~km}\) (b) \(480 \mathrm{~km}\) (c) \(360 \mathrm{~km}\) (d) none of these

Short answer

Answer: 240 km

Step-by-step solution

Find the time taken to meet each other for the first time

When P and Q are moving towards each other, their relative speed is the sum of their speeds, which is 20 km/h + 30 km/h = 50 km/h. Since they are 200 km apart, we can use the formula: Time taken to meet for the first time = Distance / Relative speed Time taken for the first meeting = \( \frac{200}{50} \) = 4 hours

Calculate the distance travelled by each person during the first meeting

We know that they met after 4 hours. We can now calculate the distance travelled by each person during this time. This can be done by multiplying their respective speeds with the time taken. Distance travelled by P = Speed * Time = 20 km/h * 4 hours = 80 km Distance travelled by Q = Speed * Time = 30 km/h * 4 hours = 120 km

Determine the remaining distance to place Y after the first meeting

Now that we know how far each person has travelled during the first meeting, we can find the remaining distance to place Y. Remaining distance to Y = Total distance - (Distance travelled by P + Distance travelled by Q) Remaining distance = 200 km - (80 km + 120 km) = 0 km As the remaining distance is 0 km, P and Q both reached Y at the same time during their first meeting.

Calculate the time and distance travelled during the second meeting

As both P and Q reached Y at the same time, they will now start moving towards X. They will meet at some point, say N, between X and Y. The time taken to meet for the second time will be the same as the first meeting (4 hours) since they are moving with the same speed. Now, we need to find how far Q has travelled during this second meeting. Again, we multiply the speed of Q by the time taken for the second meeting. Distance travelled by Q during the second meeting = Speed * Time = 30 km/h * 4 hours = 120 km

Find the total distance travelled by Q

To find the total distance travelled by Q, we have to add the distance travelled during the first meeting and the second meeting. Total distance travelled by Q = Distance travelled during the first meeting + Distance travelled during the second meeting Total distance travelled by Q = 120 km + 120 km = 240 km Thus, the distance travelled by Q when they meet each other for the second time is 240 km. The correct option is (a) 240 km.

Key concepts

Relative Speed

The idea of relative speed concerns how fast two objects are moving towards or away from each other. When two people, like P and Q, are moving towards each other, their speeds add up. This is because both are contributing to reducing the distance between them quickly.
For example, in this exercise, P and Q have speeds of 20 km/h and 30 km/h respectively. Their relative speed when moving towards each other is \(20 + 30 = 50 \) km/h.

• When moving in opposite directions, relative speed is the sum of their individual speeds.
• When moving in the same direction, relative speed is the difference between their speeds.

This is a fundamental concept to understand motion problems involving two moving objects.

Distance Calculation

Understanding how to calculate distance is crucial to solving problems in motion. Distance is essentially the length of the path traveled by a moving object during a certain period.
In this scenario, once we know the speed and time of travel, we can easily calculate the distance. The formula is:\[\text{Distance} = \text{Speed} \times \text{Time}\]For instance, Q traveled at 30 km/h for 4 hours before meeting P the first time, covering a distance of \(30 \times 4 = 120 \) km.

• This formula is used to calculate how far a person or object can travel over time.
• It applies irrespective of the direction and is fundamental for any motion problem.

Mastering this formula helps understand the movement over time and the total distance covered.

Motion Problems

Motion problems often involve calculating how long it takes for objects moving at different speeds to meet or cross each other. In this exercise, P and Q had an initial separation of 200 km.
The concept of motion applies to their journey between places X and Y. As they move back and forth, they encounter variable distances and meet at various points.

• These problems involve relative speed, distance calculations, and often multiple steps to solve them.
• It requires an understanding of initial and subsequent movements, as illustrated by the step-by-step approach following their paths and calculating their meeting times.

By decomposing the journey into these steps, they become more manageable.

Meeting Points

In the context of this exercise, meeting points are the locations where P and Q encounter each other. Their first meeting is calculated by measuring the time it takes for them to meet initially, using their relative speed and original distance.
Subsequent meetings, such as the second meeting at point N, follow a similar method. Knowing that they both turn around at place Y and head back towards X, the second meeting occurs in the same time span as the first.

• Calculating meeting points helps project when two moving entities will connect.
• These calculations rely heavily on the principles of speed, time, and distance.

Meeting points are crucial in motion exercises like these to understand how different speeds affect when and where objects will converge.