Question

two particles of mass 100 gm and 300 gm have at a given time velocity (10i -7j +6k) and( 7i- 9j+ 6k)m/s respectively the velocity of center of mass system ​

Answer 1

Velocity of center of mass of a system of two particles of masses $\sf{m_1}$ and $\sf{m_2}$ moving with velocities $\bf{v_1}$ and $\bf{v_2}$ respectively is given by,

$\longrightarrow\bf{\bar v}=\dfrac{\sf{m_1}\bf{v_1}+\sf{m_2}\bf{v_2}}{\sf{m_1+m_2}}$

According to the question,

• $\sf{m_1=100\ g}$

• $\sf{m_2=300\ g}$

• $\bf{v_1}=\sf{10}\,\bf{\hat i}-\sf{7}\,\bf{\^j}+\sf{6}\,\bf{\^k}$

• $\bf{v_2}=\sf{7}\,\bf{\hat i}-\sf{9}\,\bf{\^j}+\sf{6}\,\bf{\^k}$

Hence,

$\longrightarrow\mathbf{\bar v}=\sf{\dfrac{100\left(\sf{10}\,\bf{\hat i}-\sf{7}\,\bf{\^j}+\sf{6}\,\bf{\^k}\right)+300\left(\sf{7}\,\bf{\hat i}-\sf{9}\,\bf{\^j}+\sf{6}\,\bf{\^k}\right)}{100+300}}$

$\longrightarrow\mathbf{\bar v}=\sf{\dfrac{\sf{1000}\,\bf{\hat i}-\sf{700}\,\bf{\^j}+\sf{600}\,\bf{\^k}+\sf{2100}\,\bf{\hat i}-\sf{2700}\,\bf{\^j}+\sf{1800}\,\bf{\^k}}{400}}$

$\longrightarrow\mathbf{\bar v}=\sf{\dfrac{\sf{3100}\,\bf{\hat i}-\sf{3400}\,\bf{\^j}+\sf{2400}\,\bf{\^k}}{400}}$

$\longrightarrow\underline{\underline{\mathbf{\bar v}=\sf{\dfrac{31}{4}}\,\bf{\hat i}-\sf{\dfrac{17}{2}}\,\bf{\^j}+\sf{6}\,\bf{\^k}}}$