Question

A particle of mass m moves on the x-axis under the influence of a force of attraction towards the origin o, given by . If it starts from rest at a distance 'a' from the origin, the speed attained by it after reaching a point at a distance x will be

Answer 1

A particle of mass m moves on the x-axis under the influence of a force of attraction towards the origin o, given by $F=-\frac{k}{x^2}$

we know, from Newton's 2nd law,

F = ma => a = F/m

and also vdv/dx = a = F/m

or, vdv/dx = F/m = $-\frac{k}{mx^2}$

or, $\int\limits^v_0{v}\,dv=-\frac{k}{m}\int\limits^x_a{\frac{1}{x^2}}\,dx$

or, $\frac{v^2}{2}=-\frac{k}{m}\left[-\frac{1}{x}\right]^x_a$

or, $\frac{v^2}{2}=\frac{k}{m}\left[\frac{1}{x}-\frac{1}{a}\right]$

or, $v=\sqrt{\frac{2k(a-x)}{max}}$