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Two objects of masses 200 g and 500 g have velocities \(10\hat{i}\,\text{m/s}\) and \((3\hat{i}+5\hat{j})\,\text{m/s}\), respectively. The velocity of their centre of mass in m/s is:

1. \(5\hat{i}-25\hat{j}\)
2. \(\frac{5\hat{i}-25\hat{j}}{7}\)
3. \(\frac{25\hat{i}+5\hat{j}}{7}\)
4. \(\frac{5\hat{i}-25\hat{j}}{7}\)

Correct answer: \(\frac{25\hat{i}+5\hat{j}}{7}\)

Solution

The velocity of the centre of mass is \(\vec v_{cm}=\frac{m_1\vec v_1+m_2\vec v_2}{m_1+m_2}\). Substituting \(m_1=200\), \(m_2=500\), \(\vec v_1=10\hat i\), and \(\vec v_2=3\hat i+5\hat j\) gives \(\vec v_{cm}=\frac{2500\hat i+500\hat j}{700}=\frac{25\hat i+5\hat j}{7}\).

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