Question

Velocity of particle \(\mathrm{A}\) with respect to particle \(\mathrm{B}\) is \(4(\mathrm{~m} / \mathrm{s})\) while they are moving in same direction. And it is \(10(\mathrm{~m} / \mathrm{s})\) while they are in opposite direction. What are the velocities of the particles with respect to the stationary frame of reference. (A) \(7 \mathrm{~ms}^{-1}, 3 \mathrm{~ms}^{-1}\) (B) \(4 \mathrm{~ms}^{-1}, 5 \mathrm{~ms}^{-1}\) (C) \(7 \mathrm{~ms}^{-1}, 4 \mathrm{~ms}^{-1}\) (D) \(10 \mathrm{~ms}^{-1}, 4 \mathrm{~ms}^{-1}\)

Short answer

The velocities of particles A and B with respect to the stationary frame of reference are \(v_A = 7 \mathrm{~ms}^{-1}\) and \(v_B = 3 \mathrm{~ms}^{-1}\). So the correct answer is (A) \(7 \mathrm{~ms}^{-1}, 3 \mathrm{~ms}^{-1}\).

Step-by-step solution

Assign variables

Let the velocity of particle A with respect to a stationary frame be \(v_A\), and the velocity of particle B with respect to a stationary frame be \(v_B\).

Account for relative velocity when they are moving in the same direction

When A and B are moving in the same direction, the magnitude of relative velocity of A with respect to B is the difference in their velocities: \(v_{AB} = v_A - v_B\) When they are moving in the same direction, we know that \(v_{AB} = 4\mathrm{~m/s}\).

Account for relative velocity when they are moving in opposite directions

When A and B are moving in opposite directions, the magnitude of relative velocity of A with respect to B is the sum of their velocities: \(v_{AB} = v_A + v_B\) When they are moving in opposite directions, we know that \(v_{AB} = 10\mathrm{~m/s}\).

Set up the equations

We now have the following system of equations for the velocities of particles A and B: \(v_A - v_B = 4\) \(v_A + v_B = 10\)

Solve the system of equations

To solve this system of equations, we can use one of the many methods (like substitution, elimination, etc.). Here, we will use the elimination method: Add the two equations: \(2v_A = 14\) Now, divide by 2: \(v_A = 7 \mathrm{~ms}^{-1}\) Now that we have the value of \(v_A\), we can substitute it in either of the equations to find the value of \(v_B\). Let's substitute it in the first equation: \(7 - v_B = 4\) \(v_B = 3 \mathrm{~ms}^{-1}\)

Identify the correct option

The velocities of A and B with respect to the stationary frame of reference are: \(v_A = 7 \mathrm{~ms}^{-1}\) \(v_B = 3 \mathrm{~ms}^{-1}\) Therefore, the correct answer is: (A) \(7 \mathrm{~ms}^{-1}, 3 \mathrm{~ms}^{-1}\).

Key concepts

Frame of Reference

Understanding a frame of reference is crucial in solving problems involving relative velocity. In this context, a frame of reference is simply a point or set of points that define the position and motion of objects. A stationary frame of reference is often considered a base point, such as the ground or an observer at rest. From this perspective, we measure the velocity of particles. By using a stationary frame of reference, it is easier to calculate the absolute velocities of moving objects (particles A and B in our case), which in turn helps us to understand how they move relative to each other.

System of Equations

When solving relative velocity problems, setting up a system of equations is a powerful tool. This involves creating mathematical equations based on the given scenarios, and using these to find unknown variables, like velocities. Here, we derive two equations using the conditions of same direction and opposite direction motion:

• When moving in the same direction: \(v_A - v_B = 4\).
• When moving in opposite directions: \(v_A + v_B = 10\).

With these equations, we use methods like substitution or elimination (as in this exercise) to deduce the actual velocities of the particles. It’s a systematic approach to breaking down complex information into solvable parts.

Particle Motion

Particle motion in physics is not just about how fast a particle moves but also how it interacts in relation to other particles. For particles A and B, understanding their motion involves determining how they move within a frame of reference, and how their velocities relate to one another. Relative motion provides insight into how these particles influence each other's movement, depending on whether they are moving together or apart. This concept helps us predict and analyze the behavior of particles under different circumstances.

Directional Motion Analysis

Directional motion analysis involves examining how objects move relative to certain directions. In our example, we consider two scenarios: both particles A and B moving in the same direction and moving in opposite directions.

• Same Direction: Motion in the same direction results in a smaller relative velocity, as seen with \(v_A - v_B = 4\).
• Opposite Direction: When objects move in opposite directions, their relative velocity is larger, given by \(v_A + v_B = 10\).

This analysis is key in solving physics problems, as it breaks motion into more manageable components, allowing us to conduct thorough evaluations of each part's specific direction and speed.