Oscillations Part 1

Velocity of particle in SHM in terms of displacement x is given as
v‌‌=ω√A2−x2
⇒ v‌‌=Aω√1−x2∕A2
⇒‌

v

Aω

‌‌=√1−x2∕A2
⇒‌

v2

A2ω2

‌‌=1−‌

x2

A2

⇒‌

v2

A2ω2

+‌

x2

A2

‌‌=1 . . . (i)
As equation of ellipse is ‌

x2

a2

+‌

y2

b2

=1
Eq. (i) is similar to the equation of ellipse. So, the graph between velocity and displacement is an ellipse.