An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass 1 kg moves with a speed of $12m{s}^{-1}$ and the second part of mass 2 kg moves with $8m{s}^{-1}$ speed. If the third part flies off with $4m{s}^{-1}$ speed, then its mass is

Afte explosion

1 kg moves with speed 12 m/s (assume along $\hat{\mathrm{j}}$ )
2 kg moves with speed 8 m/s (assume along $\hat{\mathrm{i}}$ )
If the mass of other part is ‘m’ kg, it moves with speed 4 m/s.
Applying conservation of linear momentum

$\begin{array}{rl} & 12\hat{i}+16\hat{j}+m\overrightarrow{v}=0\\ & 12\hat{i}+16\hat{j}=-m\overrightarrow{v}\end{array}$

Equating magnitude on both the sides

$\begin{array}{rl} & \sqrt{{12}^2+{16}^2}=m\left(4\right)\\ & 20=m\left(4\right)\\ & m=5\mathrm{kg}\end{array}$