Question

A particle is moving on a linear pathway whose acceleration varies as `a=2x`(where, x is its distance covered) If it had started from rest(zero velocity when x=0),find an equation to suggest its velocity-displacement relationship? How to solve? Pls help

Answer 1

As the particle is moving in a linear pathway,we can say its distance covered will be equals to displacement,also because it only accelerates.

We know,accleration = rate of change in velocity = `(dv)/(dt)`

So,we can write, `(dv)/(dt) =2x`

or, `(dv)/(dx) (dx)/(dt) =v(dv)/(dx)=2x` (as, velocity = rate of change in velocity)

So,we can write,

`vdv =2xdx`

or, `int_0^V vdv =2 int_0^x xdx` (as, at `x=0,v=0`)

so, `V^2/2 = 2 (x^2/2)=x^2`

or, `V^2 =2x^2`

or, `V=sqrt(2) x`

this is the required velocity displacement relationship.