VC2.14

From Example Problems
Jump to: navigation, search

Multiply both sides of the given equation by vector 2{\frac  {dr}{dt}}\, we have

2{\frac  {dr}{dt}}\cdot {\frac  {d^{2}r}{dt^{2}}}=-2n^{2}r\cdot {\frac  {dr}{dt}}\,

Let this be Equation 1.

But {\frac  {d}{dt}}[{\frac  {dr}{dt}}]^{2}={\frac  {d}{dt}}[{\frac  {dr}{dt}}\cdot {\frac  {dr}{dt}}]={\frac  {d^{2}r}{dt^{2}}}\cdot {\frac  {dr}{dt^{2}}}+{\frac  {dr}{dt}}\cdot {\frac  {d^{2}r}{dt^{2}}}\,

Hence 2{\frac  {dr}{dt}}\cdot {\frac  {d^{2}r}{dt^{2}}}={\frac  {d}{dt}}[{\frac  {dr}{dt}}]^{2}\, [Equation 2]

Again {\frac  {d}{dt}}(r^{2})={\frac  {d}{dt}}(r\cdot r)={\frac  {dr}{dt}}\cdot r+r\cdot {\frac  {dr}{dt}}\,

Hence 2r\cdot {\frac  {dr}{dt}}={\frac  {d}{dt}}(r^{2})\, [Equation 3]

Using 2 & 3,(1) reduces to {\frac  {d}{dt}}[{\frac  {dr}{dt}}]^{2}=-n^{2}{\frac  {d}{dt}}(r^{2})\,

Integrating this,we get [{\frac  {dr}{dt}}]^{2}=-n^{2}r^{2}+c\,

Main Page