An object flying in air with velocity (20 i+25 j-12 k) suddenly breaks in two pieces whose masses are in the ratio 1: 5. The smaller mass flies off…

Let $m$m be the mass of an object flying with velocity $v$v in air. When it split into two pieces of masses in ratio $1: 5$1:5, the mass of smaller piece is $\mathrm{m} / 6$m/6 and of bigger piece is $\frac{5 \mathrm{~m}}{6}$65 m​. 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) $$\begin{aligned} & \mathrm{mv}=\frac{\mathrm{m}}{6} \mathrm{v}_1+\frac{5 \mathrm{~m}}{6} \mathrm{v}_2 \Rightarrow \mathrm{v}=\frac{\mathrm{v}_1}{6}+\frac{5 \mathrm{v}_2}{6} \\ & \text { Here, } \quad \begin{aligned} \mathrm{v} & =20 \hat{\mathrm{i}}+25 \hat{\mathrm{j}}-12 \hat{\mathrm{k}} \\ \mathrm{v}_1 & =100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}} \end{aligned} \end{aligned}$$​mv=6m​v1​+65 m​v2​⇒v=6v1​​+65v2​​ Here, vv1​​=20i^+25j^​−12k^=100i^+35j^​+8k^​​ $$\begin{array}{rlrl} \Rightarrow & 20 \hat{\mathrm{i}}+25 \hat{\mathrm{j}}-12 \hat{\mathrm{k}} & =\frac{(100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})}{6}+\frac{5 \mathrm{v}_2}{6} \\ \Rightarrow & (120 \hat{\mathrm{i}}+150 \hat{\mathrm{j}}-72 \hat{\mathrm{k}}) & =(100 \hat{\mathrm{i}}+35 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})+5 \mathrm{v}_2 \\ \Rightarrow & \mathrm{v}_2 & =\frac{1}{5}(20 \hat{\mathrm{i}}+115 \hat{\mathrm{j}}-80 \hat{\mathrm{k}}) \\ & =4 \hat{\mathrm{i}}+23 \hat{\mathrm{j}}-16 \hat{\mathrm{k}} \end{array}$$⇒⇒⇒​20i^+25j^​−12k^(120i^+150j^​−72k^)v2​=4i^+23j^​−16k^​=6(100i^+35j^​+8k^)​+65v2​​=(100i^+35j^​+8k^)+5v2​=51​(20i^+115j^​−80k^)​