Subject: Possible correction in solution.

I was solving questions on centre of mass when I encountered a problem. In the IIT questions, the first question's solution for finding the velocity of COM, they have taken the velocities as the initial velocities whereas the questions asks the velocity after 2s. Is the solution correct? If it is, then please explain.

When tackling problems related to the center of mass (COM), it's crucial to pay close attention to the details provided in the question, especially regarding time and initial conditions. In your case, it seems there might be some confusion about the velocities used in the solution. Let's break this down step by step to clarify whether the solution is indeed correct or if it needs adjustment.

Understanding the Center of Mass

The center of mass of a system is a point that represents the average position of all the mass in the system. When calculating the velocity of the center of mass, we often use the formula:

• V_COM = (m₁v₁ + m₂v₂ + ... + m_nv_n) / (m₁ + m₂ + ... + m_n)

Here, m_i represents the mass of each object, and v_i is the velocity of each object. This formula gives us the velocity of the center of mass at a specific moment in time.

Initial Velocities vs. Velocities After a Time Interval

In your question, you mentioned that the solution used initial velocities to find the velocity of the center of mass after 2 seconds. This raises an important point: if the problem asks for the velocity after a certain time, we need to consider how the velocities of the individual masses may have changed during that time.

If the objects are moving with constant velocities (i.e., no acceleration), then using the initial velocities is indeed correct. In such cases, the velocity of the center of mass remains constant, and you can simply plug in the initial velocities into the formula.

When to Use Final Velocities

However, if the objects are accelerating or if there are forces acting on them, you would need to calculate their velocities at the specific time (in this case, after 2 seconds). This involves using kinematic equations to find the final velocities:

• v = u + at

Where:

• v = final velocity
• u = initial velocity
• a = acceleration
• t = time

Example Scenario

Let’s consider a simple example. Suppose you have two masses:

• Mass 1 (m₁) = 2 kg, initial velocity (u₁) = 3 m/s, and acceleration (a₁) = 2 m/s².
• Mass 2 (m₂) = 3 kg, initial velocity (u₂) = 1 m/s, and acceleration (a₂) = 0 m/s² (constant velocity).

To find the velocities after 2 seconds:

• For Mass 1: v₁ = 3 m/s + (2 m/s² * 2 s) = 7 m/s.
• For Mass 2: v₂ = 1 m/s (since it’s constant).

Now, you can use these final velocities to calculate the velocity of the center of mass:

• V_COM = (2 kg * 7 m/s + 3 kg * 1 m/s) / (2 kg + 3 kg) = (14 + 3) / 5 = 3.4 m/s.

Final Thoughts

In summary, if the question specifically asks for the velocity after a certain time and the objects are accelerating, then the solution should indeed use the final velocities calculated after that time interval. If the objects are moving with constant velocities, then the initial velocities are appropriate. Always ensure to analyze the conditions given in the problem to determine the correct approach.