Question:

The velocity vector v and displacement vector x of a particle executing SHM are related as vdvdx=-ω²x with the initial condition V=v₀ at x=0. The velocity v, when displacement is x, is?

v=v₀²-ω²x²

v=v₀²+ω²x²

v=v₀³+ω³x³/3

V=v₀-(ω³x³/3)¹/

Correct option is A. v=√(v₀²-ω²x²)
We have vdv/dx = -ω²x
or ∫v₀ᵛvdv = -ω²∫₀ˣxdx
(v²/2 - v₀²/2) = -ω²x²/2
v² = v₀² - ω²x²
v = √(v₀² - ω²x²)