Two particles approach each other from very large distance as shown, . if the least separation between the particles is b, what is the speed of M in center of mass frame when particles are closest? (gravity negligible)

To determine the speed of particle M in the center of mass frame when two particles approach each other and reach their closest separation, we need to consider the principles of conservation of momentum and the definition of the center of mass (CM) frame. Let's break this down step by step.

Understanding the System

We have two particles, let's call them M and N, with masses m1 and m2, respectively. As they approach each other from a large distance, they will eventually reach a minimum separation distance, denoted as b. At this point, we want to find the speed of particle M in the center of mass frame.

Center of Mass Frame

The center of mass frame is a reference frame where the total momentum of the system is zero. This means that the momentum of particle M will be equal in magnitude and opposite in direction to the momentum of particle N. The position of the center of mass (CM) can be calculated using the formula:

• CM Position: x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}

In our scenario, as the particles approach each other, we can assume they are moving along a straight line towards each other. The center of mass will be located somewhere between the two particles, depending on their masses.

Conservation of Momentum

Before the particles reach their closest separation, they have some initial velocities, say v1 for particle M and v2 for particle N. The total momentum of the system before they reach the closest point can be expressed as:

• Total Momentum: P_{total} = m_1 v_1 + m_2 v_2

In the center of mass frame, the total momentum must equal zero:

• Momentum in CM Frame: m_1 v_{CM1} + m_2 v_{CM2} = 0

Finding the Speed of M

To find the speed of particle M in the center of mass frame, we can express its velocity in terms of the velocities of the particles in the lab frame. The velocity of particle M in the center of mass frame can be calculated as:

• Velocity of M in CM Frame: v_{CM1} = v_1 - v_{CM}
• Velocity of N in CM Frame: v_{CM2} = v_2 - v_{CM}

Where v_{CM} is the velocity of the center of mass, given by:

• CM Velocity: v_{CM} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}

At the point of closest approach, we can assume that the particles have not lost any kinetic energy (since gravity is negligible), and thus their speeds will be related to their masses and initial velocities. By substituting the values into our equations, we can solve for the speed of M in the center of mass frame.

Example Calculation

Let’s say particle M has a mass of 2 kg and is moving towards particle N, which has a mass of 3 kg, with initial speeds of 4 m/s and -2 m/s, respectively. First, we calculate the center of mass velocity:

• CM Velocity: v_{CM} = \frac{(2 \cdot 4) + (3 \cdot -2)}{2 + 3} = \frac{8 - 6}{5} = \frac{2}{5} = 0.4 \text{ m/s}

Now, we can find the velocity of M in the center of mass frame:

• Velocity of M in CM Frame: v_{CM1} = 4 - 0.4 = 3.6 \text{ m/s}

Thus, the speed of particle M in the center of mass frame when the particles are closest is 3.6 m/s. This approach illustrates how the conservation of momentum and the concept of the center of mass frame work together to provide insights into the motion of particles in a system.