91Ó°ÊÓ

Momentum is a fundamental concept in physics that represents the quantity of motion an object possesses. For a system of particles, total momentum can be calculated as the sum of the individual momentums. In this exercise, we explored a two-particle system, where one of the particles is stationary, while the other is in motion. Thus, the equation for total momentum becomes:

• \(P_{total} = m_1v_1 + m_2v_2\)

The given velocity of the center of mass is crucial since it helps us understand how the entire system moves as a whole. By using the concept of center of mass velocity, we found the total momentum by multiplying the combined mass by the center of mass velocity:

• \(P_{total} = Mv_{cm} = 0.40 imes 5 = 2.0 ext{ kg·m/s}\)

The momentum of the system captures how both particles, with varying states of motion, contribute to the overall movement of the system.

Velocity measures how fast and in which direction a particle is moving. It is a vector quantity composed of both speed and direction. In this scenario, one crucial step is determining the velocity of the particle located at the origin, which is initially unknown.To determine this:

• The total momentum, calculated previously as 2.0 kg·m/s, incorporates the effect of both particles.
• We know the second particle's velocity is 0 m/s since it is at rest. This simplifies the momentum equation to focus on our particle of interest.

Using the rearranged momentum equation, we solve for the velocity of the first particle:

• \(v_1 = \frac{P_{total} - m_2v_2}{m_1} = \frac{2.0 - 0}{0.30} = 6.67 ext{ m/s}\)

Thus, the velocity of the particle at the origin is 6.67 m/s, indicating it's moving considerably faster than the overall center of mass velocity.

Determining mass in a system involves understanding how it affects and is affected by motion and position. The mass calculation in this problem starts with the center of mass formula, which links the masses, respective positions, and the location of the center of mass:

• \(x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}\)

Here, with the values provided:

• The center of mass is located at 2.0 m on the x-axis.
• The first particle's position is at the origin, while the second is at 8.0 m.

To isolate and determine the unknown mass (\(m_1\)), we substituted known values into the equation, resulting in:

• \(2(m_1 + 0.10) = 0 + 0.10 imes 8\)
• Solving this equation gave us: \(m_1 = 0.30 ext{ kg}\)

Thus, the particle located at the origin has a mass of 0.30 kg. Understanding mass in this context not only helped solve for other variables but also illustrated how mass distribution affects the center of mass location.