Solve for trajectory again

Classical Mechanics Level 4

We have placed a particle of mass \(m\) at \(\overset { \rightarrow }{ r } ={ r }_{ 0 }\hat { i }\).

At \(t=0\) we have given it a velocity of \(\overrightarrow { v } ={ v }_{ 0 }\hat { j }\).In the whole co-ordinate plane the force is given by \(\overrightarrow{F}=-\overrightarrow{r}Newton\)

The title says solve for trajectory but I am telling you that the trajectory of the particle is ellipse.If the eccentricity of the ellipse is given by \(e=\frac {\sqrt {a} }{ b }\) where \(a,b\) are co-prime integers and a is square free then find \(a\times b\)

Details and Assumptions

1)\({ v }_{ 0 }=2 m/s,{ r }_{ 0 }=1 metre,m=1Kg\)

2)There are no other force acting on the particle except the force given.

3) Ignore gravity


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