Two objects of masses $200 \mathrm{~g}$ and $500 \mathrm{~g}$ possess velocities $10 \hat{\mathbf{i}}…

Two objects of masses $200 \mathrm{~g}$ and $500 \mathrm{~g}$ possess velocities $10 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}$ and $3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$ respectively. The velocity of their centre of mass in $\mathrm{m} / \mathrm{s}$ is :
  1. $5 \hat{\mathbf{i}}-25 \hat{\mathbf{j}}$
  2. $\frac{5}{7} \hat{\mathbf{i}}-25 \hat{\mathbf{j}}$
  3. $5 \hat{\mathbf{i}}+\frac{25}{7} \hat{\mathbf{j}}$
  4. $25 \hat{\mathbf{j}}-\frac{5}{7} \hat{\mathbf{j}}$

Solution

$m_1=200 \mathrm{~g}, m_2=500 \mathrm{~g}$ $v_1=10 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}, v_2=(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ velocity of centre of mass, $v_{\mathrm{CM}}=\frac{m_1 v_1+m_2 v_2}{m_1+m_2}$ $=\frac{200 \times 10 \hat{\mathbf{i}}+500(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}})}{200+500}$ $=\frac{2000 \hat{\mathbf{i}}+1500 \hat{\mathbf{i}}+2500 \hat{\mathbf{j}}}{700}$ $=\frac{3500 \hat{\mathbf{i}}+2500 \hat{\mathbf{j}}}{700}$ $=5 \hat{\mathbf{i}}+\frac{25}{7} \hat{\mathbf{j}}$

Asked in: AP EAMCET 2003

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