Problem 76 - Master Resource Book in JEE Main Physics

Question

The distance between two particles moving towards each other is decreasing at the rate of \(6 \mathrm{~m} / \mathrm{s}\). If these particles travel with same speed and in the same direction then the separatioon increase at the rate of \(4 \mathrm{~m} / \mathrm{s}\). The particles have speed as \(\begin{array}{ll}\text { (a) } 5 \mathrm{~m} / \mathrm{s} 1 \mathrm{~m} / \mathrm{s} & \text { (b) } 4 \mathrm{~m} / \mathrm{s} ; 1 \mathrm{~m} / \mathrm{s}\end{array}\) (c) \(4 \mathrm{~m} / \mathrm{s} ; 2 \mathrm{~m} / \mathrm{s}\) (d) \(5 \mathrm{~m} / \mathrm{s} ; 2 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The particles have speeds of \(5 \mathrm{~m/s}\) and \(1 \mathrm{~m/s}\), which matches option (a).

Step 1: Analyze the First Scenario

The distance between the particles decreases at a rate of \(6 \mathrm{~m/s}\). This means their speed combined in opposite directions contributes to this rate. Let the speed of each particle be \(v\). Then the equation is:\[2v = 6\]This implies that each particle moves at \(v = 3 \mathrm{~m/s}\) in one direction to account for the full \(6 \mathrm{~m/s}\) separation.

Step 2: Analyze the Second Scenario

When the particles move with the same speed and direction, the separation increases at \(4 \mathrm{~m/s}\). This scenario suggests that their speeds differ, with one leading by this amount. If they are both moving in the same direction at speed \(v_1\) and \(v_2\), we have:\[|v_1 - v_2| = 4\]This equation implies their relative speed difference leads to the \(4 \mathrm{~m/s}\) rate indicated.

Step 3: Solve for the Speeds

From the two scenarios, set two equations based on the given analysis:1. \(v_1 + v_2 = 6\)2. \(|v_1 - v_2| = 4\)We solve:Adding these two equations gives:\[2v_1 = 10 \Rightarrow v_1 = 5 \mathrm{~m/s}\] Using \(v_1 + v_2 = 6\):\[v_2 = 6 - 5 = 1 \mathrm{~m/s}\]

Step 4: Validate the Solution

Check if the calculated speeds satisfy both conditions:1. Combined speed (opposite directions) is \(6 \mathrm{~m/s}\): \(5 + 1 = 6\).2. Difference in speed (same direction) is \(4 \mathrm{~m/s}\): \(|5 - 1| = 4\).Both calculations are verified, confirming that the solution is correct.

Key Concepts

Speed Calculation

Speed is a measure of how fast something is moving. In this exercise, we're looking at two particles. When they move towards each other, their combined speed affects how quickly the distance between them changes. We know that the rate at which this distance decreases is 6 meters per second.

This means that if both particles are moving at the same speed, their speeds add up to this rate:

• Letting their speed be \( v \), and knowing their combined rate is 6 m/s, we form the equation \( 2v = 6 \).
• Solving this gives us \( v = 3 \) m/s.

Thus, each particle is moving at 3 m/s towards each other. The calculation of speed here helps us understand how the relative motion of particles affects their rate of separation.

Particle Motion

When analyzing how particles move, we must consider different scenarios of motion and how they affect the rate at which the distance between particles changes.

In this exercise, we have:

• Particles moving towards each other with identical speeds resulting in a rate of separation change: 6 m/s.
• Particles moving in the same direction at different speeds causing a change: 4 m/s in the rate of separation.

Understanding these scenarios helps us decipher how motion can either close or widen the distance between particles. When particles move towards each other, they are closing the gap. In the first scenario, while in the second scenario, moving in the same direction but at different speeds, one particle gets ahead, increasing the separation.

Rate of Change

The rate of change refers to how quickly something is changing over time. In this problem, we examine two specific rates of change related to the particles' separation:

• When moving towards one another, the rate is 6 m/s, meaning that the gap between them decreases that fast.
• When moving in the same direction, but at different speeds, the rate is 4 m/s, which means the gap increases at this speed.

To find out how their speeds are affecting these rates, we use relationships like \(|v_1 - v_2| = 4\) and \(v_1 + v_2 = 6\). Solving these equations allows us to identify the individual speeds of the particles.

By examining rates of change, students learn how different motion scenarios can impact how quickly distances alter, opening up further understanding of motion dynamics.