How do I solve this FODE for the particle's velocity?

[mr bob]

Hey.
I've been doing more mechanics recently - further kinematics in M3. I've come across another question I'm confused by.

A particle moves on the positive x-axis. When its displacement from O is x meters, is acceleration is of magnitude x^-3 m/s^2 and directed towards O. Given that initially, when t=0, the particle is at rest with x=1, find the time the particle takes to reach x=1/4.

First i integrate with respect to x to find:

V^2 = x^-2 + C where C = -1
v^2 = x^-2 -1

V = x^-1 - 1

Now i integrate again to find velocity as (dx/dt):

dx/dt = x^-1 - 1

Which is a FODE, however i can't solve it.

Thank you,
Bob

 

[LeonhardEuler]

Hey.
I've been doing more mechanics recently - further kinematics in M3. I've come across another question I'm confused by.

A particle moves on the positive x-axis. When its displacement from O is x meters, is acceleration is of magnitude x^-3 m/s^2 and directed towards O. Given that initially, when t=0, the particle is at rest with x=1, find the time the particle takes to reach x=1/4.

First i integrate with respect to x to find:

V^2 = x^-2 + C where C = -1
v^2 = x^-2 -1

V = x^-1 - 1

Now i integrate again to find velocity as (dx/dt):

dx/dt = x^-1 - 1

Which is a FODE, however i can't solve it.

Thank you,
Bob

That isn't quite right. The problem says the accleration is given by x^{-3} towards the origin, so
\frac{d^2x}{dt^2}=-x^{-3}
Multiplying by x³dt² gives:
x^3d^2x =-dt^2
Now integrate twice.

 

[HallsofIvy]

That isn't quite right. The problem says the accleration is given by x^{-3} towards the origin, so
\frac{d^2x}{dt^2}=-x^{-3}
Multiplying by x³dt² gives:
x^3d^2x =-dt^2
Now integrate twice.

No, you can't treat a second derivative like that, "differentials", dx and dy separately, are only defined for the first derivative: we could separate \frac{dv}{dt}= x^{-3} into cv= x^{-3}dt but then you can't integrate x^-3 with respect to t.

You can do this by "quadrature" (looks to me like this is what mr bob did)- since t does not appear explicitly, let v= \frac{dx}{dt}. Then \frac{d^2x}{dt^2}= \frac{dv}{dt} and, by the chain rule, \frac{dv}{dt}= \frac{dv}{dx}\frac{dx}{dt}= v\frac{dv}{dx}

That makes the equation v\frac{dv}{dx}= x^{-3} which we can separate: v dv= x^{-3}dx and, integrating, \frac{1}{2}v^2= -\frac{1}{2}x^{-2}+ C or v^2= -x^{-2}+ C almost what mr bob said (he is missing the "-" as x increases the particle slows). Taking x= 1, v= 0 when t= 0, we get 0= -1+ C so C= 1 and v^2= -x^{-2}+ C again almost as mr bob said.

However, (are you listening mr bob?) taking the square root does NOT give v= 1- x^-1! \sqrt{a^2+ b^2} \ne a+ b!

Instead v= \frac{dx}{dt}= \sqrt{1- x^{-1}}= \sqrt{\frac{x^2- 1}{x^2}}= \frac{\sqrt{x^2-1}}{x}. That separates as \frac{xdx}{x^2- 1}= dt which should be easy to integrate.

 

[HallsofIvy]

That isn't quite right. The problem says the accleration is given by x^{-3} towards the origin, so
\frac{d^2x}{dt^2}=-x^{-3}
Multiplying by x³dt² gives:
x^3d^2x =-dt^2
Now integrate twice.

No, you can't treat a second derivative like that: we could separate \frac{dv}{dt}= x^{-3} into cv= x^{-3}dt but then you can't integrat x^-3 with respect to t.

You can do this by "quadrature"- since t does not appear explicitly, let v= \frac{dx}{dt}. Then \frac{d^2x}{dt^2}= \frac{dv}{dt} and, by the chain rule, \frac{dv}{dt}= \frac{dv}{dx}\frac{dx}{dt}= v\frac{dv}{dx}<br /> <br /> That makes the equation v\frac{dv}{dx}= x^{-3} which we can separate: v dv= x^{-3}dx and, integrating, \frac{1}{2}v^2= -\frac{1}{2}x^{-2}+ C or v^2= -x^{-2}+ C <b>almost</b> what mr bob said (he is missing the &quot;-&quot; as x increases the particle slows). Taking x= 1, v= 0 when t= 0, we get 0= -1+ C so C= 1 and v^2= 1 -x^{-2} again almost as mr bob said.<br /> <br /> <b>However, (are you listening mr bob?) taking the square root does NOT give v= 1- x^-1! \sqrt{a^2+ b^2} \ne a+ b!<br /> <br /> Instead v= \frac{dx}{dt}= \sqrt{1- x^{-1}}= \sqrt{\frac{x^2- 1}{x^2}}= \frac{\sqrt{x^2-1}}{x}. That separates as \frac{xdx}{\sqrt{x^2- 1}}= dt which should be easy to integrate.</b>

 

[mr bob]

Thank you very much. I really appreciate the help. Some of these mechanics questions are a bit hard. Need to improve my integration skills abit.

 

[mr bob]

I've integrated to t = \sqrt(x^2 - 1) but can't get the answer correct. Find t when x = 1/4, which gives the root of a negative number, not real. But the book's answer is \sqrt(15) which i notice is \sqrt(x^-2 - 1). How is this possible?

Thank you,
Bob

 

[HallsofIvy]

The acceleration is of magnitude x^-3 and directed toward O!

I missed that myself: the differential equation is not
\frac{dv}{dt}= x^{-3}
it is
\frac{dv}{dt}= -x^{-3}
!

That's why you had what I (mistakenly) thought was a sign error!

 

[mr bob]

When i integrate i get t = 1 - (1 - x^2).
I'm getting confused by this now, i think my brain's just died, ah well. Sorry for keeping asking questions, i can see how annoying it would be if the answer was so obvious yet i don't grasp it.

Thanks,
Bob