To solve the problem of finding the velocity of the center of mass of two particles, we can follow these steps:
### Step 1: Identify the masses and velocities of the particles
- Let \( m_1 = 3 \, \text{kg} \) (moving due north with velocity \( v_1 = 2 \, \text{m/s} \))
- Let \( m_2 = 2 \, \text{kg} \) (moving due east with velocity \( v_2 = 3 \, \text{m/s} \))
### Step 2: Define the coordinate system
- We will use a Cartesian coordinate system where:
- North direction corresponds to the positive y-axis (j-cap)
- East direction corresponds to the positive x-axis (i-cap)
### Step 3: Write the velocities in vector form
- The velocity of the first particle (3 kg) can be represented as:
\[
\vec{v_1} = 0 \, \hat{i} + 2 \, \hat{j} \, \text{m/s}
\]
- The velocity of the second particle (2 kg) can be represented as:
\[
\vec{v_2} = 3 \, \hat{i} + 0 \, \hat{j} \, \text{m/s}
\]
### Step 4: Calculate the velocity of the center of mass
The formula for the velocity of the center of mass \( \vec{v_{cm}} \) is given by:
\[
\vec{v_{cm}} = \frac{m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1 + m_2}
\]
Substituting the values:
\[
\vec{v_{cm}} = \frac{3 \, (0 \, \hat{i} + 2 \, \hat{j}) + 2 \, (3 \, \hat{i} + 0 \, \hat{j})}{3 + 2}
\]
Calculating the numerator:
\[
= \frac{(0 + 6) \, \hat{i} + (6) \, \hat{j}}{5} = \frac{6 \, \hat{i} + 6 \, \hat{j}}{5} = \frac{6}{5} \, \hat{i} + \frac{6}{5} \, \hat{j}
\]
### Step 5: Simplify the expression
Thus, the velocity of the center of mass is:
\[
\vec{v_{cm}} = \frac{6}{5} \, \hat{i} + \frac{6}{5} \, \hat{j} \, \text{m/s}
\]
### Step 6: Calculate the magnitude of the velocity of the center of mass
The magnitude \( v_{cm} \) can be calculated using the Pythagorean theorem:
\[
v_{cm} = \sqrt{\left(\frac{6}{5}\right)^2 + \left(\frac{6}{5}\right)^2} = \sqrt{2 \left(\frac{6}{5}\right)^2} = \frac{6}{5} \sqrt{2}
\]
### Step 7: Determine the direction of the velocity
The direction can be found using the tangent function:
\[
\tan(\theta) = \frac{v_y}{v_x} = \frac{\frac{6}{5}}{\frac{6}{5}} = 1
\]
Thus, \( \theta = 45^\circ \) which indicates that the direction is northeast.
### Final Result
The magnitude of the velocity of the center of mass is:
\[
v_{cm} = \frac{6}{5} \sqrt{2} \, \text{m/s} \approx 1.2 \sqrt{2} \, \text{m/s}
\]
And the direction is northeast.
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