Question

Two objects are moving along the y-axis. Object 1 has a mass of 5 kg and a velocity of 3 m/s. Object 2 has a mass of 10 kg and a velocity of 2 m/s. What is their center of mass?

Answer 1

To find the center of mass of the two objects moving along the y-axis, we can use the formula for the center of mass \(y_{cm}\) of a system of particles:

\[
y_{cm} = \frac{\sum (m_i y_i)}{\sum m_i}
\]

where \(m_i\) is the mass of each object and \(y_i\) is the position of each object along the y-axis. In this case, we can assume the initial positions of the objects:

- Let \(y_1\) be the position of Object 1 (5 kg).
- Let \(y_2\) be the position of Object 2 (10 kg).

Since you didn’t provide specific positions, we will denote the positions of the objects as follows:
- Let the position of Object 1, \(y_1\), be 0 m.
- Let the position of Object 2, \(y_2\), be at an arbitrary position (let's call it \(a\) meters).

Now plugging these into the center of mass equation:

\[
y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2}
\]
\[
y_{cm} = \frac{(5 \, \text{kg} \cdot 0 \, \text{m}) + (10 \, \text{kg} \cdot a)}{5 \, \text{kg} + 10 \, \text{kg}}
\]
\[
y_{cm} = \frac{0 + 10a}{15}
\]
\[
y_{cm} = \frac{10a}{15} = \frac{2a}{3}
\]

The center of mass is located at \(\frac{2a}{3}\) on the y-axis.

If you provide specific positions for Objects 1 and 2, I can calculate the exact center of mass for you!