What Determines the Relative Velocity of Two Attracting Masses?

[zorro]

Homework Statement

2 particles of masses M and m are initially at rest and infinitely separated. When they move towards each other due to gravitational attraction, what is their relative velocity at any instant?

Homework Equations

The Attempt at a Solution

The problem may be treated in centre of mass system co-ordinates. Therefore,
reduced mass R= Mm/(M+m)

Kinetic energy = 0.5RV_rel²

Work done in moving masses from infinite distance to a separation distance d can be found by integration.
W.D. = -GMm/d

Now by using Work-Energy theorem,
K.E._f - K.E._i = W.D

0.5RV_rel² - 0 = -GMm/d

V_rel = [-2G(M+m)/d]^1/2
but the answer is
[2G(M+m)/d]^1/2

Where am I wrong?

 

[rl.bhat]

Using Work-Energy theorem, you can write
K.E.f - K.E.i = W.D = P.E.i - P.E.f

 

[zorro]

Thanks!

 

[D H]

V_rel = [-2G(M+m)/d]^1/2

Where am I wrong?

<rant>
It's always a good idea to do a sanity check on your answer. Check your units, check your signs, create a somewhat simpler problem that should give approximately the correct answer, etc. You won't always have an answer book against which you can compare your result. Get in the habit of double-checking everything and you won't be lost when the answer book disappears.
</rant>

In this problem, G, M, m, and d are all positive quantities. That means your radical is negative, and that in turn means your relative velocity is imaginary. Now does an imaginary velocity make a bit of sense here? It does not of course, so that means you did something wrong.

Your mistake is a sign error in work. Work is just one of those things you just have to be careful about with respect to sign. Is positive work work that is done by the system or work that is done on the system? I can't tell you which one is right because both schemes are used.

 

[zorro]

<rant>
It's always a good idea to do a sanity check on your answer. Check your units, check your signs, create a somewhat simpler problem that should give approximately the correct answer, etc. You won't always have an answer book against which you can compare your result. Get in the habit of double-checking everything and you won't be lost when the answer book disappears.
</rant>

Thanks for your advise. I will keep that in mind.

Is positive work work that is done by the system or work that is done on the system? I can't tell you which one is right because both schemes are used.

I know that the sign scheme for work done on the system (+ve) or by the system (-ve) is applied in the case of thermodynamics (gases). I never knew we could apply that rule here also.