Question

Two particles \(\mathrm{P}\) and \(\mathrm{Q}\) get \(5 \mathrm{~m}\) closer each second while travelling in opposite direction. They get \(1 \mathrm{~m}\) closer each second while travelling in same direction. The speeds of \(\mathrm{P}\) and \(\mathrm{Q}\) are respectively (A) \(5 \mathrm{~ms}^{-1}, 1 \mathrm{~ms}^{-1}\) (B) \(3 \mathrm{~ms}^{-1}, 4 \mathrm{~ms}^{-1}\) (C) \(3 \mathrm{~ms}^{-1}, 2 \mathrm{~ms}^{-1}\) (D) \(10 \mathrm{~ms}^{-1}, 5 \mathrm{~ms}^{-1}\)

Short answer

The speeds of particles P and Q are \(3 \mathrm{~ms}^{-1}\) and \(2 \mathrm{~ms}^{-1}\), respectively (option C).

Step-by-step solution

Setting up equations for the relative speeds

Let's denote the speed of particle P as \(v_P\) and the speed of particle Q as \(v_Q\). When the particles are moving in opposite directions, they get 5 m closer each second. When they are moving in the same direction, they get 1 m closer each second. We can set up two equations using this information: 1. When particles are moving in opposite directions: \(v_P + v_Q = 5\) 2. When particles are moving in the same direction: \(v_P - v_Q = 1\)

Solving the system of equations simultaneously

Let us solve these equations to find the values for \(v_P\) and \(v_Q\). We can eliminate one of the variables by adding both equations: \vspace{2mm} \( (v_P + v_Q) + (v_P - v_Q) = 5 + 1 \) This simplifies to: \vspace{2mm} \(2v_P = 6\) Now, we can find the value for \(v_P\): \vspace{2mm} \(v_P = \frac{6}{2} = 3 \mathrm{~ms}^{-1}\) Next, we can substitute the value of \(v_P\) into one of the initial equations to find \(v_Q\). Let's use the first equation: \vspace{2mm} \(v_Q = 5 - v_P = 5 - 3 = 2 \mathrm{~ms}^{-1}\)

Match the answer with the given options

We have found the speeds of particles P and Q to be \(3 \mathrm{~ms}^{-1}\) and \(2 \mathrm{~ms}^{-1}\), respectively. This matches with option (C): (C) \(3 \mathrm{~ms}^{-1}, 2 \mathrm{~ms}^{-1}\)

Key concepts

Linear Motion

Linear motion occurs when an object moves along a straight path in either direction. It is one of the simplest forms of motion to understand and analyze. In the context of our exercise, the particles P and Q are moving linearly, which means they are following a path in one dimension, either towards or away from each other.

Key aspects of linear motion include:

• Speed: This is the rate at which an object covers distance. For linear motion, speed remains consistent unless acted upon by external forces.
• Direction: In a linear motion problem, the direction determines whether velocities add up or subtract from each other.

Understanding linear motion helps students develop a foundational grasp of more complex physics topics. It simplifies the analysis as it deals with one-dimensional movement, making calculations and system analysis more straightforward.

Systems of Equations

When deciphering problems involving multiple elements, like particles moving at different speeds, systems of equations become invaluable. They allow us to solve for unknowns by setting up relationships between variables.

In the exercise, two main equations were established based on the particles' relative motion:

• When particles are traveling in opposite directions, their speeds sum up to cause the closing movement: \(v_P + v_Q = 5\).
• Conversely, when they move in the same direction, the difference in their speeds results in a slower closing rate: \(v_P - v_Q = 1\).

By solving these simultaneous equations, students learn to find exact values for the variables in question. This technique is a powerful tool in both mathematics and physics, offering a clearer view of diverse problems, whether they involve traffic flow, economic models, or particle dynamics, like our example above.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion. It focuses on parameters such as displacement, velocity, and time.

In the problem involving particles P and Q, understanding kinematics allows us to predict and describe their positions over time, given their velocities. Here are some fundamental elements:

• Displacement: The change in position of particles P and Q as they get closer or further apart.
• Velocity: In our specific context, the velocities of P and Q are crucial to understanding how quickly they close or increase the distance between them.
• Time: This is typically set as one second in many problems to evaluate momentary changes.

By mastering kinematics, students can dissect complex motion scenarios into more digestible components, focusing on how objects move through time and space in practical applications.