Problem 79 - Master Resource Book in JEE Main Physics

Question

Two cars \(A\) and \(B\) are moving with same speed of \(45 \mathrm{~km} / \mathrm{h}\) along same direction. If a third car \(\mathrm{C}\) coming from the opposite direction with a speed of \(36 \mathrm{~km} / \mathrm{h}\) meets two cars in an interval of 5 minutes. The distance between cars \(A\) and \(B\) should be (in \(\mathrm{km}\) ) (a) \(6.75\) (b) \(7.25\) (c) \(5.55\) (d) \(8.35\)

Step-by-Step Solution

Answer

The distance between cars A and B is 6.75 km (option a).

Step 1: Understand the Problem

We are given that two cars, A and B, move in the same direction at 45 km/h, and car C moves in the opposite direction at 36 km/h. Car C meets A and then meets B after 5 minutes. We need to find the distance between A and B.

Step 2: Convert Time to Hours

Since the time interval is given in minutes, convert 5 minutes into hours: \[ 5 \text{ minutes} = \frac{5}{60} \text{ hours} = \frac{1}{12} \text{ hours} \]

Step 3: Calculate Relative Speed

Calculate the relative speed between car C and cars A and B. Since they are moving in opposite directions, add their speeds:\[ \text{Relative speed} = 45 \text{ km/h} + 36 \text{ km/h} = 81 \text{ km/h} \]

Step 4: Calculate Distance Travelled by Car C

Using the relative speed, calculate the distance that car C travels to meet car B after meeting car A, using the formula \(\text{Distance} = \text{Speed} \times \text{Time}\):\[ \text{Distance} = 81 \text{ km/h} \times \frac{1}{12} \text{ hours} = 6.75 \text{ km} \]

Step 5: Select the Correct Answer

The distance that car C travels to meet car B after meeting car A represents the distance between cars A and B. Thus, the correct answer is 6.75 km.

Key Concepts

Kinematics

Kinematics is a branch of physics concerned with the motion of objects without considering the forces that cause this motion. It helps us describe, analyze, and predict how objects move. In this exercise, you're exploring a situation where multiple moving objects (cars in this case) are involved. To solve related problems, it is vital to understand key concepts such as velocity, which is the speed of something in a given direction.

When cars A and B are moving in the same direction and car C is coming from the opposite direction, we focus on the relative velocity concept. Relative velocity refers to how fast one object is moving in relation to another object. By knowing the speeds of the cars and the direction they are moving, we can determine how quickly the distance between them changes.

In this case, since car C is moving towards cars A and B, you add their respective speeds to find the relative velocity between them. This lays the groundwork for calculating how long it will take car C to meet each car.

Speed and Distance

Speed and distance are fundamental concepts in physics that help describe motion. **Speed** is defined as the distance traveled per unit of time and helps determine how fast an object is moving. **Distance**, on the other hand, is the total movement of an object without regard to direction.

In this exercise, the speeds of the cars and the time period provide useful data to find the distance between cars A and B. It's crucial to convert time into appropriate units, which in this exercise is converting minutes to hours, to ensure consistency when calculating distance using the speed-time relation:

• Distance = Speed × Time

By using the relative speed calculated during kinematics, you can find out how far car C travels between meeting cars A and B. This distance travelled is essentially the gap between the two cars.

Problem Solving in Physics

Problem solving in physics often involves multiple steps, logical reasoning, and application of core principles. This exercise demonstrates a classic physics problem that requires converting time units, calculating relative speeds, and using formulas appropriately.

Here's a step-by-step summary to solve such problems:

• **Understand the problem**: Clearly identify what's given and what needs to be found.
• **Convert units if needed**: Consistent units are essential to simplify calculations.
• **Use relevant formulas**: Identify and use correct physics formulas like Distance = Speed × Time.
• **Calculate and check**: Ensure calculations are logical and make sense in the context.
• **Select the right answer**: Compare your result with the options (if any) provided and choose the most plausible one.

Applying these steps systematically helps solve complex physics problems and can greatly enhance understanding and retention of concepts.