# If the three particles of masses m1, m2, and m3 are moving with velocity v1, v2, and v3 respectively, then the velocity of the center of mass is.

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If the three particles of masses m1, m2, and m3 are moving with velocity v1, v2, and v3 respectively, then the velocity of the center of mass is.
1. $\frac{m_1v_1+m_2v_2+m_3v_3}{m_1+m_2+m_3}$
2. $\frac{v_1+v_2+v_3}{m_1+m_2+m_3}$
3. $\frac{m_1v_1^2+m_2v_2^2+m_3v_3^2}{m_1v_1+m_2v_2+m_3v_3}$
4. None of these

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Correct Answer - Option 1 : $\frac{m_1v_1+m_2v_2+m_3v_3}{m_1+m_2+m_3}$

CONCEPT:

Centre of mass:

• The centre of mass of a body or system of a particle is defined as, a point at which the whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.

The motion of the centre of mass:

• Let there are n particles of masses m1, m2,..., mn.
• If all the masses are moving then,

⇒ Mv = m1v1 + m2v2 + ... + mnvn

⇒ Ma = m1a1 + m2a2 + ... + mnan

⇒ $M\vec{a}=\vec{F_1}+\vec{F_2}+...+\vec{F_n}$

⇒ M = m1 + m2 + ... + mn

• Thus, the total mass of a system of particles times the acceleration of its centre of mass is the vector sum of all the forces acting on the system of particles.
• The internal forces contribute nothing to the motion of the centre of mass.

EXPLANATION:

• We know that if a system of particles have n particles and all are moving with some velocity, then the velocity of the centre of mass is given as,

⇒ $V=\frac{m_1v_1+m_2v_2+...+m_nv_n}{m_1+m_2+...+m_n}$     -----(1)

• Therefore if the three particles of masses m1, m2, and m3 are moving with velocity v1, v2, and v3 respectively, then the velocity of the centre of mass is given as,

⇒ $V=\frac{m_1v_1+m_2v_2+m_3v_3}{m_1+m_2+m_3}$

• Hence, option 1 is correct.