Question

QUESTION: 2 Two particles having the same mass M.
moving towards each other with velocity 4v and 2v
respectively. What is the velocity of the centre of mass.​

Answer 1

The velocity of center of mass, about a certain point, of a system of two particles of masses $\displaystyle\sf {m_1}$ and $\displaystyle\sf {m_2}$ having velocities $\displaystyle\sf {v_1}$ and $\displaystyle\sf {v_2}$ is given by,

$\displaystyle\longrightarrow\sf{\bar v=\dfrac {m_1v_1+m_2v_2}{m_1+m_2}}$

The concept is based on law of conservation of linear momentum.

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Here the mass of the two particles are same.

• $\displaystyle\sf {m_1=m_2=M}$

The velocity of the first particle is,

• $\displaystyle\sf {v_1=4v}$

Both particles are moving towards each other so the second particle moves opposite to the first particle. Hence the velocity of the second particle is,

• $\displaystyle\sf {v_2=-2v}$

Now the velocity of the center of mass is,

$\displaystyle\longrightarrow\sf{\bar v=\dfrac {m_1v_1+m_2v_2}{m_1+m_2}}$

$\displaystyle\longrightarrow\sf{\bar v=\dfrac {M(4v)+M(-2v)}{M+M}}$

$\displaystyle\longrightarrow\sf{\bar v=\dfrac {4Mv-2Mv}{2M}}$

$\displaystyle\longrightarrow\sf{\bar v=\dfrac {2Mv}{2M}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{\bar v=v}}}$

Hence the velocity of the center of mass is v in the direction of the first particle.