Finding position and velocity from Force (analytic mechanics)

  • #1

Homework Statement


A body of mass "m" is repelled from the origin by a force F(x). The body is at rest at x_0, a distance from the origin, at t=0. Find v(x) and x(t).


Homework Equations


[tex]F(x)=\frac{k}{x^3}[/tex]
[tex]\ddot{x}=\frac{d\dot{x}}{dt}=\frac{dx}{dt}\frac{d\dot{x}}{dx}=v\frac{dv}{dx}[/tex]


The Attempt at a Solution



I first perform the following steps:

[tex]F=ma=m\ddot{x}=\frac{k}{x^3}[/tex]
[tex]\int{v{dv}}=\int_{x_0}^{x}\frac{k}{mx^3}dx[/tex]
[tex]\frac{v^2}{2}=\frac{-k}{2mx^2}+\frac{k}{2m{x_0}^2}[/tex]
[tex]v(x)=\sqrt{\frac{k}{m}(x_{0}^{-2}-x^{-2})}[/tex]

The next step is to integrate v(x) WRT time. Doing this will solve for the position. My homework states that x(t) should be:

[tex]x(t)=\sqrt{\frac{kt^2}{mx_{0}^{2}}+x_{0}^{2}}[/tex]

If I integrate v(x) WRT time, I will get a different answer for position than the equation above. Is my expression for v(x) incorrect?
 

Answers and Replies

  • #2
nasu
Gold Member
3,785
438
You should integrate v(t) over time and not v(x).
As you don't have v(t) you can instead write
dx/dt=v(x) , separate the variables and integrate both sides.
I mean, integrate
dx/v(x)= dt

You'll get t = some function of x. Then solve for x and you'll get x(t).
 

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