Two identical masses, each \(1\text{ }\text{kg}\), having velocity vectors:\(\vec{{\mathrm{V}}_{\mathrm{A}}}={\alpha }_{…

\(\vec{{\mathrm{V}}_{\mathrm{A}}}={\alpha }_{1}{\mathrm{t}}^{2}\overset{^}{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\overset{^}{\mathrm{j}}+{\alpha }_{3}\overset{^}{\mathrm{k}},\) \(\vec{{\mathrm{V}}_{\mathrm{B}}}={\alpha }_{1}\overset{^}{\mathrm{i}}+{\alpha }_{2}\overset{^}{\mathrm{j}}+{\alpha }_{3}{\mathrm{t}}^{2}\overset{^}{\mathrm{k}},\) \({\alpha }_{1}=2,\text{ }{\alpha }_{2}=3n,\text{ }{\alpha }_{3}=4p\)

\(\mathrm{For}\mathrm{the}\mathrm{velocities}\mathrm{to}\mathrm{be}\mathrm{orthogonal}:\\ \vec{{\mathrm{V}}_{\mathrm{A}}}.\vec{{\mathrm{V}}_{\mathrm{B}}}=0\\ ({\alpha }_{1}{\mathrm{t}}^{2}){\alpha }_{1}+({\alpha }_{2}\mathrm{t}){\alpha }_{2}+({\alpha }_{3})({\alpha }_{3}{\mathrm{t}}^{2})=0\\ \mathrm{At},\mathrm{t}=1\\ {\alpha }_{1}^{2}+{\alpha }_{2}^{2}+{\alpha }_{3}^{2}=0\\ \mathrm{Substitute}{\alpha }_{1}=2,{\alpha }_{2}=3\mathrm{n},{\alpha }_{3}=4\mathrm{p}\\ {2}^{2}+(3\mathrm{n}{)}^{2}+(4\mathrm{p}{)}^{2}=0\\ 4+9{\mathrm{n}}^{2}+16{\mathrm{p}}^{2}=0....(1)\)

\(\vec{{\mathrm{r}}_{\mathrm{A}}}=\int \vec{{\mathrm{V}}_{\mathrm{A}}}\mathrm{dt}=\int ({\alpha }_{1}{\mathrm{t}}^{2}\overset{^}{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\overset{^}{\mathrm{j}}+{\alpha }_{3}\overset{^}{\mathrm{k}})\mathrm{dt}={\alpha }_{1}\frac{{\mathrm{t}}^{3}}{3}\overset{^}{\mathrm{i}}+{\alpha }_{2}\frac{{\mathrm{t}}^{2}}{2}\overset{^}{\mathrm{j}}+{\alpha }_{3}\mathrm{t}\overset{^}{\mathrm{k}}\\ \vec{{\mathrm{r}}_{\mathrm{B}}}=\int \vec{{\mathrm{V}}_{\mathrm{B}}}\mathrm{dt}=\int ({\alpha }_{1}\overset{^}{\mathrm{i}}+{\alpha }_{2}\overset{^}{\mathrm{j}}+{\alpha }_{3}{\mathrm{t}}^{2}\overset{^}{\mathrm{k}})\mathrm{dt}={\alpha }_{1}\mathrm{t}\overset{^}{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\overset{^}{\mathrm{j}}+{\alpha }_{3}\frac{{\mathrm{t}}^{3}}{3}\overset{^}{\mathrm{k}}\\ \mathrm{At}\mathrm{t}=1\mathrm{s}\\ \vec{{\mathrm{r}}_{\mathrm{A}}}=\frac{{\alpha }_{1}}{3}\overset{^}{\mathrm{i}}+\frac{{\alpha }_{2}}{2}\overset{^}{\mathrm{j}}+{\alpha }_{3}\overset{^}{\mathrm{k}}\\ \vec{{\mathrm{r}}_{\mathrm{B}}}={\alpha }_{1}\overset{^}{\mathrm{i}}+{\alpha }_{2}\overset{^}{\mathrm{j}}+\frac{{\alpha }_{3}}{3}\overset{^}{\mathrm{k}}\)

\(\mathrm{Relative}\mathrm{displacement}:\\ r{⃗}_{AB}=r{⃗}_{A}-r{⃗}_{B}=\left(\frac{{\alpha }_{1}}{3}-{\alpha }_{1}\right)\overset{^}{\mathrm{i}}+\left(\frac{{\alpha }_{2}}{2}-{\alpha }_{2}\right)\overset{^}{\mathrm{j}}+\left({\alpha }_{3}-\frac{{\alpha }_{3}}{3}\right)\overset{^}{\mathrm{k}}\\ r{⃗}_{AB}=\left(-\frac{2{\alpha }_{1}}{3}\right)\overset{^}{\mathrm{i}}+\left(-\frac{{\alpha }_{2}}{2}\right)\overset{^}{\mathrm{j}}+\left(\frac{2{\alpha }_{3}}{3}\right)\overset{^}{\mathrm{k}}\\ \mathrm{Substitute}{\alpha }_{1}=2,{\alpha }_{2}=3\mathrm{n},{\alpha }_{3}=4\mathrm{p}\\ r{⃗}_{AB}=\left(-\frac{4}{3}\right)\overset{^}{\mathrm{i}}+\left(-\frac{3\mathrm{n}}{2}\right)\overset{^}{\mathrm{j}}+\left(\frac{8\mathrm{p}}{3}\right)\overset{^}{\mathrm{k}}\)

Relative velocity is given by

\(\vec{{\mathrm{V}}_{\mathrm{AB}}}=\vec{{\mathrm{V}}_{\mathrm{A}}}-\vec{{\mathrm{V}}_{\mathrm{B}}}=\\ \vec{{\mathrm{V}}_{\mathrm{AB}}}={\alpha }_{1}{\mathrm{t}}^{2}\overset{^}{\mathrm{i}}+{\alpha }_{2}\mathrm{t}\overset{^}{\mathrm{j}}+{\alpha }_{3}\overset{^}{\mathrm{k}}-{\alpha }_{1}\overset{^}{\mathrm{i}}-{\alpha }_{2}\overset{^}{\mathrm{j}}-{\alpha }_{3}{\mathrm{t}}^{2}\overset{^}{\mathrm{k}}\\ \mathrm{At}\mathrm{t}=1\mathrm{s}\\ \vec{{\mathrm{V}}_{\mathrm{AB}}}=0\overset{^}{\mathrm{i}}+0\overset{^}{\mathrm{j}}+0\overset{^}{\mathrm{k}}\)

The angular momentum of A with respect to B s givne by

\(\vec{\mathrm{L}}=\mathrm{m}\left(\vec{{\mathrm{r}}_{\mathrm{AB}}}\times \vec{{\mathrm{V}}_{\mathrm{AB}}}\right)=\vec{0}\mathrm{As}\vec{{\mathrm{V}}_{\mathrm{AB}}}=0\)