Question

A stone of mass 30 kg after exclusion it is divided into two equal parts . prove that both stone moves in opposite direction.

Answer 1

Assume the stone was initially at rest. After exploding into two, let the mass of one part be $\displaystyle\sf {m}$ (in kg) so that the mass of the other will be $\displaystyle\sf {30-m.}$

Let the first part (of mass $\displaystyle\sf {m)}$ attain a velocity $\displaystyle\bf {v_1,}$ and the other part, $\displaystyle\bf {v_2,}$ after explosion.

Then, by linear momentum conservation,

$\displaystyle\longrightarrow\sf{m}\bf{v_1}\ +\ \sf{(30-m)}\bf{v_2}=\sf{0}$

since both were at rest initially.

Then,

$\displaystyle\longrightarrow\sf{(30-m)}\bf{v_2}\ =\ -\sf{m}\bf{v_1}$

$\displaystyle\longrightarrow\bf{v_2}\ =\ -\sf{\left (\dfrac {m}{30-m}\right)}\bf{v_1}\sf{\quad\quad\dots (1)}$

Since $\displaystyle\sf {m\ \textless\ 30,\ \dfrac {m}{30-m}\ \textgreater\ 0.}$

Then, from (1) we get,

$\displaystyle\longrightarrow\underline {\underline{\bf{v_2}\ \propto\ -\bf{v_1}}}$

This implies the velocity of first part is opposite to that of the other, and hence both parts move opposite to each other.

Hence the Proof!