A projectile following the usual parabolic trajectory explodes into fragments midway in the air due to internal forces. The centre of mass of the system will

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 particles appeared to be concentrated.

The motion of the centre of mass:

Let there be 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.

Thus the internal forces contribute nothing to the motion of the centre of mass.

Explanation:

The body explodes somewhere during the flight and splits into a number of particles.

When the body explodes only internal forces act on the body and we know that the internal forces contribute nothing to the motion of the centre of mass.

Therefore the centre of mass will follow the same parabolic path after the explosion.