The velocity v and displacement x of a particle executing simple harmonic motion are related as
At x=0,v=v0. Find the velocity v when the displacement becomes x.
We have, vdvdx=−ω2x
or, v dv=−ω2x dx
or, v∫v0v dv=x∫0−ω2x dx .......(i)
When summation is made on −ω2 x dx the quantity to be varied is x. When summation is made on vdv the quantity to be varied is v. As x varies from 0 to x the velocity varies from v0 to v. Therefore, on the left the limits of integration are from v0 to v and on the right they are from 0 to x. Simplifying (i),