The velocity v and displacement x of a particle executing simple harmonic motion are related as
`v (dv)/(dx)= -omega^2 x`.
`At x=0, v=v_0.` Find the velocity v when the displacement becomes x.

Explanation

We have
`v(dv)/(dx)=-omega^2x
or, `v dv=-omega^2x dx`
or, `int^v _v_0 v dv = int^x_0-omega^2 x dx`…..i
When summation is made on `-omega^2 x dx` the quantity to be varied is x. When summation is made on v dv the quantity to be varied is v. As x varies from 0 to x the velocity varies from `v_0 to v`. Thereforre, on teh left the limits of integration are from `v_0 to v` and on the right they are from 0 to x. Simplifying i.
`[1/2 v^2]^v_v_0=-omega^2[x^2/2]^x_0`
or ` 1/2(v^2-v^2_0)=-omega^2 x^2/2`
or, `v^2=v^2_0-omega^2x^2`
or `v= sqrt(v^2_0-omega^2 x^2)`