Question

Two objects of masses 2 kg and 5 kg possess velocities \( 10 \hat{i} \, {m/s} \) and \( 3 \hat{i} + 5 \hat{j} \, {m/s} \) respectively. Then the velocity of C.M. in m/s is:

• A. \( 5 \hat{i} - 25 \hat{j} \)
• B. \( 5 \hat{i} + \frac{25}{7} \hat{j} \)
• C. \( \frac{5}{7} \hat{i} - 25 \hat{j} \)
• D. \( 25 \hat{i} - \frac{5}{7} \hat{j} \)

Hint:

For systems of particles, the velocity of the center of mass is the weighted average of the velocities of the particles, with masses as the weights.

Answer 1

The Correct Option is B

The velocity of the center of mass is given by the formula: \[ \vec{v}_{{CM}} = \frac{m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1 + m_2} \] Substitute the values: \[ \vec{v}_{{CM}} = \frac{(2)(10 \hat{i}) + (5)(3 \hat{i} + 5 \hat{j})}{2 + 5} \] \[ \vec{v}_{{CM}} = \frac{20 \hat{i} + 15 \hat{i} + 25 \hat{j}}{7} \] \[ \vec{v}_{{CM}} = \frac{35 \hat{i} + 25 \hat{j}}{7} \] \[ \vec{v}_{{CM}} = \left(\frac{35}{7}\right) \hat{i} + \left(\frac{25}{7}\right) \hat{j} \] \[ \vec{v}_{{CM}} = 5 \hat{i} + \frac{25}{7} \hat{j} \, {m/s}. \] Thus, the velocity of the center of mass is \( 5 \hat{i} + \frac{25}{7} \hat{j} \, {m/s} \).