Mechanics (Application of Integration)

[Muddyakka]

A particle of mass 1 kg moves in a straight line from the positive x - axis towards the origin. The particle starts from rest with displacement p meters. When its displacement from the origin is x meters, it experiences a force of -k/x² Newtons.

(a) Prove that the velocity-displacement relationship is given by [MATH]-sqrt(2k/p)((p-x)/x)[/MATH]
(b) Hence, find the time required for the particle to reach the origin.

I am stuck here.I am not sure if what I have is correct and if it is how I should continue.

The answer is πsqrt(p³/8k)

 

[Muddyakka]

I know that I have to do a trig sub but I am not sure which one.
I am also not sure how to simplify the P once I do a trig sub.

 

[skeeter]

[MATH]\frac{dx}{\sqrt{\frac{p}{x}-1}} = -\sqrt{\frac{2k}{p}} \, dt[/MATH]
I used the substitution [MATH]x=p\cos^2{\phi}[/MATH]
however, after trying twice, I arrived at [MATH]x(t)=0[/MATH] at [MATH]t=\sqrt{\frac{p^3}{2k}}[/MATH] instead of the given solution you posted.

I may have made a mistake somewhere ... maybe someone else can confirm (or deny) the validity of my solution.

 

[Muddyakka]

Thanks for this! I tried the same sub and everything looks fine. Perhaps the answer key was wrong....