Question

If a particle of mass \(5.0 \mathrm{kg}\) is moving east at $10 \mathrm{m} / \mathrm{s}\( and a particle of mass \)15 \mathrm{kg}\( is moving west at \)10 \mathrm{m} / \mathrm{s},$ what is the velocity of the CM of the pair?

Short answer

In a two-particle system with masses \(m_1 = 5.0 \ kg\) and \(m_2 = 15 \ kg\), and velocities \(v_1 = 10 \ m/s\) east and \(v_2 = -10 \ m/s\) west, the velocity of the center of mass is \(-5 \ m/s\). This indicates that the center of mass is moving westward at a speed of \(5 \ m/s\).

Step-by-step solution

Note the given values

We are given: - Mass of particle 1, \(m_1 = 5.0 \mathrm{kg}\) - Velocity of particle 1, \(v_1 = 10 \mathrm{m/s}\) (east) - Mass of particle 2, \(m_2 = 15 \mathrm{kg}\) - Velocity of particle 2, \(v_2 = -10 \mathrm{m/s}\) (west)

Determine the sum of particle masses

Calculate the total mass of the system by adding the masses of the two particles: $$m_{total} = m_1 + m_2 = 5.0 \mathrm{kg} + 15 \mathrm{kg} = 20 \mathrm{kg}$$

Calculate the weighted sum of particle velocities

Next, we'll calculate the weighted sum of the particles' velocities using their masses: $$m_1 v_1 + m_2 v_2 = (5.0 \mathrm{kg})(10 \mathrm{m/s}) + (15 \mathrm{kg})(-10 \mathrm{m/s}) = 50 \mathrm{kg \cdot m/s} - 150 \mathrm{kg \cdot m/s} = -100 \mathrm{kg \cdot m/s}$$

Calculate the velocity of the center of mass

Now, we'll use the formula for the velocity of the center of mass and plug in our values for the weighted sum of velocities and the total mass: $$v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} = \frac{-100 \mathrm{kg \cdot m/s}}{20 \mathrm{kg}} = -5 \mathrm{m/s}$$

Interpret the result

The velocity of the center of mass of the two particles is \(-5 \mathrm{m/s}\). Since it is negative, the center of mass is moving westward at a speed of \(5 \mathrm{m/s}\).