An object flying in air with velocity $(20\hat{\mathrm{i}} + 25\hat{\mathrm{j}} - 12\hat{\mathrm{k}})$ suddenly breaks in two pieces whose masses are in the ratio 1: 5. The smaller mass flies off with a velocity $(100\hat{\mathrm{i}} + 35\hat{\mathrm{j}} + 8\hat{\mathrm{k}})$. The velocity of the larger piece will be

Solution:

Let m be the mass of an object flying with velocity v in air. When it gets split into two pieces of masses in ratio 1 : 5, the mass of smaller piece is m/6 and of bigger piece is $\frac{5\mathrm{m}}{6}.$

This situation can be interpreted diagrammatically as below.

As, the object breaks in two pieces, so the momentum of the system will remains conserved, i.e. the total momentum (before breaking) = total momentum (after breaking)

                     $m\mathbf{v}=\frac{m}{6}{\mathbf{v}}_1+\frac{5m}{6}{\mathbf{v}}_2\Rightarrow \mathbf{v}=\frac{{\mathbf{v}}_1}{6}+\frac{5{\mathbf{v}}_2}{6}$

Given,  $v= 20\hat{\mathrm{i}} + 25\hat{\mathrm{j}} - 12\hat{\mathrm{k}}$

and      $v_1 = 100\hat{\mathrm{i}} + 35\hat{\mathrm{j}} + 8\hat{\mathrm{k}}$

Putting these values in Eq. (i), we get

$(20\hat{\mathbf{i}}+25\hat{\mathbf{j}}-12\hat{\mathbf{k}})=\frac{(100\hat{\mathbf{i}}+35\hat{\mathbf{j}}+8\hat{\mathbf{k}})}{6}+\frac{5{\mathbf{v}}_2}{6}$

$\Rightarrow (120\hat{\mathbf{i}}+150\hat{\mathbf{j}}-72\hat{\mathbf{k}})=(100\hat{\mathbf{i}}+35\hat{\mathbf{j}}+8\hat{\mathbf{k}})+5{\mathbf{v}}_2$

$\Rightarrow {\mathbf{v}}_2=\frac{1}{5}(20\hat{\mathbf{i}}+115\hat{\mathbf{j}}-80\hat{\mathbf{k}})$

                   $=4\hat{\mathrm{i}} + 23\hat{\mathrm{j}} - 16\hat{\mathrm{k}}$