The ratio of the velocities of the two cars A and B is 2 : 1. Both the cars are moving towards each other. If the centre of mass of both the cars is not moving then find the ratio of the mass of the car A to car B.

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 = 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.

CALCULATION:

Given vA = 2v, vB = -v, and V = 0 m/sec

• Both the cars are moving in the opposite direction so if the velocity of car A is positive then the velocity of car B will be negative.
• 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,

⇒        -----(1)

By equation 1 the velocity of the centre of mass of the cars is given as,

⇒ 

⇒ 

⇒ 2v × m_A = v × m_B

⇒ 

• Hence, option 2 is correct.