Two bodies of mass $4 \mathrm{~kg}$ and $5 \mathrm{~kg}$ are moving along east and north directions with…

Two bodies of mass $4 \mathrm{~kg}$ and $5 \mathrm{~kg}$ are moving along east and north directions with velocities $5 \mathrm{~m} / \mathrm{s}$ and $3 \mathrm{~m} / \mathrm{s}$ respectively. Magnitude of the velocity of centre of mass of the system is
  1. $\frac{25}{9} \mathrm{~m} / \mathrm{s}$
  2. $\frac{9}{25} \mathrm{~m} / \mathrm{s}$
  3. $\frac{41}{9} \mathrm{~m} / \mathrm{s}$
  4. $\frac{16}{9} \mathrm{~m} / \mathrm{s}$

Solution

Velocity of centre of mass is $ \mathbf{v}_{\mathrm{CM}}=\frac{m_1 \mathbf{v}_1+m_2 \mathbf{v}_2}{m_1+m_2} $ Given, $m_1=4 \mathrm{~kg}, m_2=5 \mathrm{~kg}, v_1=5 \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$ $ \begin{aligned} & v_2=3 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s} \\ & \therefore \quad \mathbf{v}_{\mathrm{CM}}=\frac{4 \times 5 \hat{\mathbf{j}}+5 \times 3 \hat{\mathbf{i}}}{5+4} \\ &=\frac{20 \hat{\mathbf{i}}}{9}+\frac{15}{9} \hat{\mathbf{j}} \end{aligned} $ Hence, magnitude $\left|\mathbf{v}_{\mathrm{CM}}\right|=\sqrt{\left(\frac{20}{9}\right)^2+\left(\frac{15}{9}\right)^2}$ $ =\frac{25}{9} \mathrm{~m} / \mathrm{s} $

Asked in: AP EAMCET 2013

Practice more Center of Mass Momentum and Collision questions on Aicharya