Question

A body of mass 1 kg, initially at rest, explodes and breaks into three fragments of masses in the ratio $1:1:3$. The two pieces of equal mass fly off perpendicular to each other with a speed of $30m/s$ each. If the velocity of the heavier fragment is $10\sqrt{x}$, determine x.

Answer 1

Total mass of the body $M=1kg$

So, mass of first fragment $m_1=\frac{1}{5}\times 1=0.2kg$

Mass of second fragment $m_2=\frac{1}{5}\times 1=0.2kg$

Mass of third fragment $m_3=\frac{3}{5}\times 1=0.6kg$

Velocity of masses $m_1$ and $m_2$, $v=30m/s$

Momentum of first fragment $p_1=m_{1}v=0.2\times 30=6kgm/s$

Similarly, $p_2=m_{2}v=0.2\times 30=6kgm/s$

Third fragment will fly in a direction opposite to the direction of resultant of $p_1$ and $p_2$ so that total momentum of the system after explosion remains zero.

Momentum of the third fragment $p_3=\sqrt{p_1^2+p_2^2}$

$\therefore$ $p_3=\sqrt{6^2+6^2}=6\sqrt{2}kgm/s$

Thus speed of third fragment $v_3=\frac{p_3}{m_3}=\frac{6\sqrt{2}}{0.6}=10\sqrt{2}m/s$

$⟹x=2$