The velocities of three particles of masses $20 \mathrm{~g}$, $30 \mathrm{~g}$ and $50 \mathrm{~g}$ are $10…

The velocities of three particles of masses $20 \mathrm{~g}$, $30 \mathrm{~g}$ and $50 \mathrm{~g}$ are $10 \hat{\mathbf{i}}, 10 \hat{\mathbf{j}}$ and $10 \hat{\mathbf{k}}$, respectively. The velocity of the centre of mass of the three particles is
  1. $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \mathbf{k}$
  2. $10(\mathbf{i}+\mathbf{j}+\mathbf{k})$
  3. $20 \hat{i}+30 \hat{\mathbf{j}}+5 \mathbf{k}$
  4. $2 \hat{\mathbf{i}}+30 \hat{\mathbf{j}}+50 \hat{\mathbf{k}}$

Solution

$\begin{aligned} & m_1=20 \mathrm{~g}, m_2=30 \mathrm{~g}, m_3=50 \mathrm{~g} \\ & \mathbf{v}_1=10 \hat{\mathbf{i}}, \mathbf{v}_2=10 \hat{\mathbf{j}}, \mathbf{v}_3=10 \hat{\mathbf{k}}\end{aligned}$ $\begin{aligned} & v_{\text {cm }}=\frac{m_1 \mathbf{v}_1+m_2 \mathbf{v}_2+m_3 \mathbf{v}_3}{m_1+m_2+m_3} \\ &=\frac{20 \times 10^{-3} \times 10 \hat{\mathbf{i}}+30 \times 10^{-3} \times 10 \hat{\mathbf{i}}+50 \times 10^{-3} \times 10 \hat{\mathbf{k}}}{(20+30+50) \times 10^{-3}} \\ &=\frac{200 \hat{\mathbf{i}}+300 \hat{\mathbf{j}}+500 \hat{\mathbf{k}}}{100} \\ &=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\end{aligned}$

Asked in: AP EAMCET 2001

Practice more Center of Mass Momentum and Collision questions on Aicharya