Question

Two masses, \(m_{1}=2.0 \mathrm{~kg}\) and \(m_{2}=3.0 \mathrm{~kg}\), are moving in the \(x y\) -plane. The velocity of their center of mass and the velocity of mass 1 relative to mass 2 are given by the vectors \(v_{\mathrm{cm}}=(-1.0,+2.4) \mathrm{m} / \mathrm{s}\) and \(v_{\mathrm{rel}}=(+5.0,+1.0) \mathrm{m} / \mathrm{s} .\) Determine a) the total momentum of the system b) the momentum of mass 1 , and c) the momentum of mass 2 .

Short answer

The total momentum of the system is (-5.0, +12.0) kg.m/s. b) What is the momentum of mass 1? The momentum of mass 1 is (+6.0, -4.0) kg.m/s. c) What is the momentum of mass 2? The momentum of mass 2 is (-11.0, +16.0) kg.m/s.

Step-by-step solution

a) Total momentum of the system

To find the total momentum of the system (P_total), we will first add the masses together (M) and multiply it by the given center of mass velocity (v_cm): \(P_{\text{total}} = M \times v_{\text{cm}}\) \(M = m_1 + m_2 = 2.0 kg + 3.0 kg = 5.0 kg\) \(v_{\text{cm}} = (-1.0,+2.4) \frac{m}{s}\) So, \(P_{\text{total}} = (5.0 kg) \times (-1.0,+2.4) \frac{m}{s} = (-5.0,+12.0) \frac{kg.m}{s}\)

b) Momentum of mass 1

To find the momentum of mass 1 (P_1), we need to first find its velocity (v_1). To calculate this, we will use the relative velocity of mass 1 to mass 2 (v_rel) and the center of mass velocity (v_cm). We know that: \(v_{\text{rel}} = v_1 - v_2\) \(v_1 = v_{\text{rel}} + v_2\) Now we can find the velocity of mass 2 (v_2) using the center of mass velocity: \(v_{\text{cm}} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}\) \(v_2 = \frac{v_{\text{cm}}(m_1 + m_2) - m_1 v_1}{m_2}\) Substitute the values: \(v_2 = \frac{(-1.0,+2.4)(5.0) - 2.0 v_1}{3.0}\) \(v_2 = \frac{(-5.0,+12.0) - 2.0 v_1}{3.0}\) Now, substitute \(v_2\) back into the equation for \(v_1\): \(v_1 = (+5.0,+1.0) + \frac{(-5.0,+12.0) - 2.0 v_1}{3.0}\) Solving for \(v_1\), we get: \(v_1 = (+3.0,-2.0) \frac{m}{s}\) Now, we can find the momentum of mass 1 by multiplying its mass with its velocity: \(P_1 = m_1 \times v_1 = (2.0 kg) \times (+3.0,-2.0) \frac{m}{s} = (+6.0,-4.0) \frac{kg.m}{s}\)

c) Momentum of mass 2

Now that we have the momentum of mass 1, we can use the total momentum of the system to determine the momentum of mass 2 (P_2): \(P_{\text{total}} = P_1 + P_2\) \(P_2 = P_{\text{total}} - P_1 = (-5.0,+12.0) \frac{kg.m}{s} - (+6.0,-4.0) \frac{kg.m}{s} = (-11.0,+16.0) \frac{kg.m}{s}\)

Key concepts

Total Momentum

Total momentum is a critical concept in physics, often referring to the sum of momenta of all the objects in a system. In this exercise, the total momentum of the system is found by first determining the combined mass of the objects, and then using the center of mass velocity. The equation used is:

• \( P_{\text{total}} = M \times v_{\text{cm}} \)

where \( M \) is the sum of \( m_1 \) and \( m_2 \). For our example, their total mass \( M \) is 5.0 kg. The center of mass velocity \( v_{\text{cm}} \) is given as \((-1.0, +2.4)\,\frac{m}{s}\). By multiplying, we find:
\((5.0 \text{ kg}) \times (-1.0, +2.4)\,\frac{m}{s} = (-5.0, +12.0)\,\frac{kg \cdot m}{s}\)
This shows that the total momentum encompasses both linear motion and direction, providing insight into how the system as a whole is moving.

Relative Velocity

Relative velocity is a vector quantity that shows how fast one object is moving in relation to another. This exercise highlights its importance in understanding the momentum of individual components. The relative velocity \( v_{\text{rel}} \) between mass 1 and mass 2 is given as \((+5.0, +1.0) \,\frac{m}{s}\). This helps determine the velocity of one mass with respect to another:

• \( v_{1} = v_{\text{rel}} + v_{2} \)

Finding \( v_2 \) involves the center of mass velocity equation, solving, and substituting back gives \( v_1 = (+3.0, -2.0) \,\frac{m}{s}\).
Understanding relative velocity lets students grasp how each mass moves in relation to the other, key in many physics problems involving multiple objects.

Momentum of Mass Patterns

The "momentum of mass patterns" refers to understanding how each individual object's momentum contributes to the system's overall behavior. Calculating the momentum of individual masses helps illustrate this concept. For mass 1, we use its velocity \( v_{1} = (+3.0, -2.0) \,\frac{m}{s}\) and multiply by its mass:

• \( P_1 = m_1 \times v_1 = (2.0 \text{ kg}) \times (+3.0, -2.0) \,\frac{m}{s} = (+6.0, -4.0) \,\frac{kg \cdot m}{s} \)

Mass 2's momentum \( P_2 \) is derived by knowing the total momentum and subtracting \( P_1 \):
\( P_2 = P_{\text{total}} - P_1 = (-5.0, +12.0) \,\frac{kg \cdot m}{s} - (+6.0, -4.0) \,\frac{kg \cdot m}{s} = (-11.0, +16.0) \,\frac{kg \cdot m}{s} \)
Recognizing these patterns helps students visualize how each mass's movement contributes to the whole system, a key stepping stone in mastering dynamics in physics.