Rectilinear motion of two attracting masses

[H. Leone]

Homework Statement

Consider two masses of variable magnitude (M m) that are separated by a distance ( r ). Both masses are free to move. Calculate dr/dt.

Homework Equations

(See below)

The Attempt at a Solution

F = GMm / r²
a = Gm / r²
let k = Gm
da/dr = -2kr^-3
dr/da = (-2k)^-1 r³
dr/da * da/dt = dr/dt

Just need to solve for da/dt and I don't know how to do that.

 

[haruspex]

Are you sure you have stated the question correctly? I would expect it to ask for d²r/dt². There is no way to know dr/dt without further information.
The acceleration you found, Gm/r², is only for mass M.

 

[H. Leone]

Are you sure you have stated the question correctly? I would expect it to ask for d²r/dt². There is no way to know dr/dt without further information.
The acceleration you found, Gm/r², is only for mass M.

As far as stating the equation, I honestly can't say if I've done it correctly. The question is a matter of personal interest and didn't come from any textbook. I apologise for that. I just wanted to find a relationship between the time elapsed and the distance between two unbound masses. I find it really peculiar how you can solve for d²r/dt² and not dr/dt. Why is that exactly? You said you needed more information. Would it help at all if we said that both masses were stationary at t=0?

Thanks for taking the time to read :)

 

[haruspex]

I find it really peculiar how you can solve for d2r/dt2 and not dr/dt.

You can solve for dr/dt. The simplest is to find dr/dt as a function of r. You can use conservation of energy for that. Or you can derive it from the force=mass*acceleration equation using (d/dt)(v) = (dr/dt)(d/dr)(v) = v dv/dr. Finding dr/dt as a function of t (or r as a function of t) is harder.

You said you needed more information. Would it help at all if we said that both masses were stationary at t=0?

Yes and no. Your original formulation asked for dr/dt as a function of r. Knowing dr/dt at a given t does not help since we don't what what r is at that time. If you specify dr/dt at some initial r then you can find dr/dt as a function of r. If you want to know r or dr/dt as a function of t then you need an initial r and an initial t.

 

[H. Leone]

You can solve for dr/dt. The simplest is to find dr/dt as a function of r. You can use conservation of energy for that. Or you can derive it from the force=mass*acceleration equation using (d/dt)(v) = (dr/dt)(d/dr)(v) = v dv/dr. Finding dr/dt as a function of t (or r as a function of t) is harder.

Yes and no. Your original formulation asked for dr/dt as a function of r. Knowing dr/dt at a given t does not help since we don't what what r is at that time. If you specify dr/dt at some initial r then you can find dr/dt as a function of r. If you want to know r or dr/dt as a function of t then you need an initial r and an initial t.

Oh neat! I hadn't considered conservation of energy. So something like this?

1/2 mv² = -GMm / r
I'm cancelling out the mass (m) out from both sides because I can, but I don't know if I should.
1/2 v² = -GM/r
let k = -GM
v = (2k/r)^0.5
Where v is presumably dr/dt? In your second method you seem to distinguish between dr/dt and v, leading me to think that I'm incorrect here.

Also, a question regarding syntax: how is "dr/dt as a function of r," different from "dr/dt as a function of t?"

 

[haruspex]

Oh neat! I hadn't considered conservation of energy. So something like this?
1/2 mv² = -GMm / r

no, that's saying KE=PE. Conservation says the total is constant (need not be zero, or even positive).

Where v is presumably dr/dt? In your second method you seem to distinguish between dr/dt and v, leading me to think that I'm incorrect here.

No, dr/dt is the same as v. Just used v to make the notation easier.

Also, a question regarding syntax: how is "dr/dt as a function of r," different from "dr/dt as a function of t?"

dr/dt = sin(r) would be a differential equation with dr/dt as a function of r; dr/dt=kt would be one with dr/dt as a function of t. In the present problem, you can readily find out how dr/dt depends on r, but it is much harder to find how it depends on t.

 

[H. Leone]

no, that's saying KE=PE. Conservation says the total is constant (need not be zero, or even positive).

No, dr/dt is the same as v. Just used v to make the notation easier.

dr/dt = sin(r) would be a differential equation with dr/dt as a function of r; dr/dt=kt would be one with dr/dt as a function of t. In the present problem, you can readily find out how dr/dt depends on r, but it is much harder to find how it depends on t.

Oh, right. I don't know what I was thinking. So for the conservation of energy, that equation would look more like:

1/2 mv₁² + 1/2 Mv₂² - GMm/ r = c (constant)
(I'm including the kinetic energies of both masses because gravitational potential energy is written in terms of M and m )
The problem now is that we have two velocities to describe the change in r.
Would I be correct in saying that abs (v₁) + abs (v₂) = dr/dt ?

In the future, I may try to find dr/dt as a function of t. What would you consider to be a good starting point for that?

 

[theodoros.mihos]

Would I be correct in saying that abs (v1) + abs (v2) = dr/dt ?

Yes. This equation can produced by time derivation of

1/2 mv12 + 1/2 Mv22 - GMm/ r = c (constant)

 

[haruspex]

Yes. This equation can produced by time derivation of

I don't see how.

The problem now is that we have two velocities to describe the change in r.
Would I be correct in saying that abs (v1) + abs (v2) = dr/dt ?

Not quite. dr/dt can be negative, and the two velocities could have opposite signs. That equation covers neither of those cases.

 

[theodoros.mihos]

I don't see how.

$x_1$, $x_2$, $r$ are functions of time.

 

[haruspex]

$x_1$, $x_2$, $r$ are functions of time.

Sure, but if you differentiate the energy equation m, M and G are not going to disappear. So how will you deduce an equation that contains none of those? The relationship between the two velocities and dr/dt is purely kinematic. It owes nothing to forces.

 

[theodoros.mihos]

Disapear because $F_1+F_2=0$ by 3o Newton's Law.

 

[haruspex]

Disapear because $F_1+F_2=0$ by 3o Newton's Law.

The relationship between the two velocities and dr/dt has nothing to do with Newton's Laws. It is a simple fact of geometry and kinematics. You could change the gravitational law (and, consistently, energy conservation etc.) and the relationship would still hold. If you have a way of obtaining that relationship by differentiating the energy equation please post it, preferably in a private mail, not on this thread. I can imagine a way, but it goes something like this:
- differentiate the energy equation with respect to distance, so a force equation
- Apply N3 so that what you should end up with is 0=0, but along the way use kinematics to feed in the relationship between the velocities and the distance, and
- voila, all you have left is that kinematic relationship

 

[H. Leone]

This is fascinating stuff. It's pretty clear to me that I've got a lot more to learn in way of calculus and mechanics.
Thank you all for the responses!

 

[rcgldr]

The two objects accelerate towards a common center of mass. Let m1 be on the left accelerating to the right, and m2 on the right accelerating to the left.

a1 = F/m1 = (G m1 m2 / r) / m1 = G m2 / r²
a2 = F/m1 = -( G m1 m2 / r) / m2 = - G m1 / r²
a2 = a1 (a2 / a1) = (-m1/m2)a1
v2 = v1 (v2 / v1) = (-m1/m2)v1
let v = rate of closure = v2 - v1 = (-m1/m2)v1 - v1 = - (1 + m1/m2)v1 = -((m1 + m2)/m2)v1
v1 = -m2/(m1+m2) v
v2 = -(m1/m2) v1 = -(m1/m2)(-m2/(m1+m2)) v = m1/(m1+m2) v
Let r0 = initial distance:
1/2 m1 v1² + 1/2 m2 v2² = G m1 m2 / r - G m1 m2 / r0 = G m1 m2 (r0 - r)/(r0 r) =
1/2 m1 (v m2/(m1+m2))² + 1/2 m2 (v m1/(m1+m2))² = G m1 m2 (r0 - r)/(r0 r)
1/2 (v/(m1 + m2))² ((m1 m2²) + (m1² m2)) = G m1 m2 (r0 - r)/(r0 r)
1/2 (v/(m1 + m2))² (m1 + m2) = G (r0 - r)/(r0 r)
1/2 v² / (m1 + m2) = G (r0 - r)/(r0 r)
1/2 v² = G (m1 + m2) (r0 - r)/(r0 r)
v = - sqrt(2 G (m1 + m2) / r0) sqrt((r0 - r) / r)

Alternate method:
let a = d(dr/dt)/dt = a2 - a1
a = -G (m1 + m2 ) / r²
use chain rule to get to v as a function of r based on v(r(t))
a = dv/dt = (dv/dr)(dr/dt) = v dv/dr
v dv/dr = -G (m1 + m2 ) / r²
v dv = -G (m1 + m2 ) dr / r²
integrate and use - G (m1 + m2 ) / r0 as constant
1/2 v^2 = G (m1 + m2 ) / r - G (m1 + m2 ) / r0 = G (m1 + m2 ) (r0 - r)/(r0 r)
v = - sqrt(2 G (m1 + m2) / r0) sqrt((r0 - r) / r)

Contuning from here leads to a tricky integral:
v = dr/dt = -sqrt(2 G (m1 + m2) / r0) sqrt((r0 - r) / r)
$- \ \sqrt{\frac{r_0}{2 G (m1 + m1)}} \sqrt{\frac{r}{r_0-r}} dr = dt$
one method is to define θ as
$r = r_0 \ sin^2(θ)$
$dr = 2\ r_0 \ sin(θ)\ cos(θ) dθ$
$θ = {sin^{-1}\left ( \sqrt{\frac{r}{r_0}} \right )}$
$\sqrt{\frac{r}{r_0-r}} = \sqrt{\frac{r_0 \ sin^2(θ)}{r_0-r_0 \ sin^2(θ)}} = \sqrt{\frac{r_0 \ sin^2(θ)}{r_0(1\ -\ sin^2(θ))}} = \sqrt{\frac{r_0 \ sin^2(θ)}{r_0\ cos^2(θ)}} = tan(θ)$
resulting in
$-\sqrt{\frac{r_0}{2 \ G \ (m1 + m1)}} \ (2 \ r_0 \ sin^2(θ)) dθ = dt$
$t = -\frac{{r_0}^{3/2}}{\sqrt{2 \ G \ (m1 + m2)}} \int_{θ_0}^{θ_1}2 \ sin^2(θ) dθ = -\frac{{r_0}^{3/2}}{\sqrt{2 \ G \ (m1 +m2)}} \left [ θ \ -\ sin(θ)cos(θ) \right ]_{θ_0}^{θ_1}$