Rocket that explodes mid-trajectory

Problem

A rocket in ballistic flight explodes mid-trajectory, splitting into two fragments. Explain, using Newton’s second law for the centre of mass, why (in the absence of air resistance) the centre of mass of the fragments continues to follow the original parabolic trajectory. State what changes in the CM frame immediately after the explosion.

Solution

Why the parabola remains. The explosion is due to forces internal to the system: by Newton’s third law they occur in equal and opposite pairs and cancel in the sum. The only external force remaining is weight. Hence $\sum \vec{F}_\text{ext} = M_\text{tot}\,\vec{a}_\text{CM} = M_\text{tot}\,\vec{g}$ The centre of mass continues to have acceleration $\vec{g}$, exactly as before: it carries on undisturbed along the same parabola, as if the explosion had not happened.

In the CM frame. Immediately after the explosion, in the CM frame the total momentum is zero ($v_\text{CM}$ is unchanged by the internal collision). The two fragments therefore set off with equal and opposite momenta: $m_1\vec{v}_1' = -m_2\vec{v}_2'$. Viewed from the CM, the explosion appears perfectly symmetric, with the pieces moving apart in opposite directions.

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Topics: Centre of mass Concepts: Centre of mass · Newton’s second law · Explosion