Question

 A particle of mass m moves under action of force F = – kr (k is positive constant). Find r(t) and v(t). (Boundary conditions r(0) = r_o, v(0) = v_o, at t=0). 

Answer 1

Given,

Mass of the particle = m

Action of force $(F)=-kr$

$a=\frac{F}{m}$

Now,

$a=\frac{-kr}{m}$

We know that acceleration $a=\frac{dv}{dt}=\frac{dv dr}{dr dt}=v\frac{dv}{dr}$

Now, substituting the values,

$\Rightarrow \frac{vdv}{dr}=\frac{-kr}{m}$

$\Rightarrow vdv= \frac{-kr}{m}dr$

Now, taking the integration of both side of the given equation,

$\Rightarrow \int_ {v_1=0}^{v_2=v} vdv = \int_{r_1=0}^{r_2=r} \frac{-kr}{m}dr$

$\Rightarrow \frac{v^2}{2}=\frac{-kr^2}{2m}$