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Question:

The velocity vector v and displacement vector x of a particle executing SHM are related as vdv/dx = ω²x with the initial condition v = v₀ at x = 0. The velocity v, when displacement is x, is:

v=3√(v₀³+ω³x³)

v=√(v₀²+ω²x²)

v=√(v₀²-ω²x²)

v=v₀-(ω³x³eˣ)¹/³

Solution:

As it is SHM so the equation of motion will be F = -kx or vdv/dx = -ω²x
Now integrating the expression with boundary condition,
∫(v₀→v)vdv = -ω²∫(₀→x)xdx
or 1/2[v² - v₀²] = -ω²x²/2
or v = √(v₀² - ω²x²)