Calculating Speed of Stationary Object Falling Towards Earth

[RandallB]

For an object put into position say halfway to the moon.
Take away all orbital and any other speed – that is it’s completely stationary with respect to the earth. Of course it will immediately start to freefall directly to earth.
Acceleration is increasing so calculating the speed at different points cannot use a simple acceleration formula.

Is there a straight forward formula that gives its speed as a function of distance that accounts for the increasing acceleration?

RB

 

[pervect]

For an object put into position say halfway to the moon.
Take away all orbital and any other speed – that is it’s completely stationary with respect to the earth. Of course it will immediately start to freefall directly to earth.
Acceleration is increasing so calculating the speed at different points cannot use a simple acceleration formula.
Is there a straight forward formula that gives its speed as a function of distance that accounts for the increasing acceleration?
RB

This is just a particular example of the general orbital problem - I assume you are assuming there is no air resistance, and are working in an inertial frame (not a frame rotating along with the Earth)

Halfway to the moon, the moon's gravity is only 1/80 th of the Earth's gravity, so it is still reasonable to negelect the moon's gravity.

The easy way to solve the problem of an orbit around a single body is too note that there are two conserved quantites, energy and angular momentum. If the angular momentum is zero, as it would be for a body falling straight into the Earth, the problem is even easier

Energy = m(-GM/r+ .5 m v_r^2 + .5 m v_theta^2)
Angular Momentum = m r v_theta

The fact that angular momentum is zero means v_theta is zero, thus the velocity of the object at every point is strictly radial (directly towards the planet) and the solution of the first equation gives v = v_r as a function of r.

If for some reason the angular momentum isn't zero then you need to fingure out v_theta first from the value of Angular momentum, and then compute v_r from the first formula. The total velocity v will be v= sqrt(v_r^2 + v_theta^2).

 

[RandallB]

Energy = m(-GM/r+ .5 m v_r^2 + .5 m v_theta^2)
Angular Momentum = m r v_theta
The fact that angular momentum is zero means v_theta is zero, thus the velocity of the object at every point is strictly radial (directly towards the planet) and the solution of the first equation
gives v = v_r as a function of r.

??
v = v_r
I don’t understand that – maybe it’s the font’s
Or I just don’t know how to read it
Is r radius ??
What is v_ ?

If V_max = Max speed
When object were allowed to fall to center of Earth core & all that mass at that single center point

Do you mean r = (r_max - r_current )

So r would be zero when object at limit and at a max when object crosses center core of earth?

Then “v_” = V_max/ r_max

But that seems to liner.

I must be missing something in the text fonts.

 

[Janus]

Okay,
The total energy of the object can be expressed as
$$E_t = \frac{mv^2}{2} - \frac{GMm}{r}$$
m is the mass of the object
M the mass of the Earth
v the velocity of the obect
r is the distance from the center of the Earth.
At its start the object's velocity is zero, so we get
$$E_t =- \frac{GMm}{r_{int}}$$
where R_int is the initial starting distance from the Earth.
Energy is conserved so
$$- \frac{GMm}{r_{int}} = \frac{mv^2}{2}- \frac{GMm}{r}$$

$$- \frac{GM}{r_{int}} = \frac{v^2}{2}- \frac{GM}{r}$$

$$\frac{GM}{r}-\frac{GM}{r_{int}} = \frac{v^2}{2}$$

$$2GM \left(\frac{1}{r}-\frac{1}{r_{int}} \right) = v^2$$

$$\sqrt{2GM \left(\frac{1}{r}-\frac{1}{r_{int}} \right)} = v$$

Since the new radius r is equal to the distance traveled minus the distance traveled:

$$r= r_{int}-d$$

then:

$$\sqrt{2GM \left(\frac{1}{r_{int}-d}-\frac{1}{r_{int}} \right)} = v$$

which gives you the velocity after falling a distance d from a height of r_int

 

[pervect]

v_r is the radial component of the velocity, the component directed towards the Earth.

v_theta is the angular component of the velocity, perpendicular to v_r.

Earth...x

at point 'x', v_r points to the left, v_theta points up

If I understand your question correctly, v_theta is zero, and thus v=v_r.

You then just substitute v=v_r in the first equation (the second equation becomes irrelevant) to find v as a function of r.

Janus's post was much more complete, hopefully he explains it all in more detail and more clearly than I did.

 

[RandallB]

$$\sqrt{2GM \left(\frac{1}{r}-\frac{1}{r_{int}} \right)} = v$$

$$r= r_{int}-d$$

$$\sqrt{2GM \left(\frac{1}{r_{int}-d}-\frac{1}{r_{int}} \right)} = v$$

which gives you the velocity after falling a distance d from a height of r_int

This doesn’t seem to make sense to me.
The implication is the speed at the Earth center should go to the sqrt of infinity.

I think your working with conserving the energy as measured from the distant infinity. That is the energy being conserved gets smaller at greater r.
Thus it looks like your solving for the escape speed which does get smaller at greater r.

Here were looking at an increasing potential as the farther out we start the more energy we have to convert to higher speeds as we approach the center of the gravitational sorce.

Maybe I need to work on replacing the g with GM/r² in the gravity formula somehow solving for r or d.

 

[pervect]

The formula is good only when the object never gets closer to the center of the Earth than the Earth's radius - i.e. that objects stop when they hit the Earth's surface.

If we let r_int = infinity, then the formula reduces to the formula for escape velocity as per the wikipedia

http://en.wikipedia.org/wiki/Escape_velocity

(with the appropriate change of variable).

You might also want to review the wikipedia (or a physics text) about gravitational potential energy - the formula follows directly and simply from the conservation of energy.

http://en.wikipedia.org/wiki/Gravitational_potential

 

[Janus]

This doesn’t seem to make sense to me.
The implication is the speed at the Earth center should go to the sqrt of infinity.

You have to remember that this equation works for all values of r, if you treat the Earth as a point mass, and and as long as r is greater that the radius of the Earth, it is perfectly okay to do so. So it works right up to the point where your falling object hits the surface of the Earth. If you want to assume that you have also drilled a hole in the Earth so that the object continues to fall after it passes the Earth's surface, you will need to use a different equation to calculate its further acceleration.

I think your working with conserving the energy as measured from the distant infinity. That is the energy being conserved gets smaller at greater r.
Thus it looks like your solving for the escape speed which does get smaller at greater r.

It doesn't matter where you measured the energy from (where you put zero potential energy) as it is the differential in potential energy between your starting point and the point a distance D towards the Earth that causes the increase in velocity and this remains the same no matter where you measured it from

Here were looking at an increasing potential as the farther out we start the more energy we have to convert to higher speeds as we approach the center of the gravitational source.

The potential is increasing with the equation I gave.
$\frac{Mmv^2}{2}$ is the kinetic energy and
$- \frac{GMm}{r}$ is the gravitational potential energy
Note.as r increases, the equation becomes less negative and the potential energy increases.

$$E_t = \frac{Mmv^2}{2}- \frac{GMm}{r}$$
Simply means that the total energy is equal to the kinetic energy plus the gravitation potential energy.

Maybe I need to work on replacing the g with GM/r² in the gravity formula somehow solving for r or d.

 

[RandallB]

You have to remember that this equation works for all values of r, if you treat the Earth as a point mass, and and as long as r is greater that the radius of the Earth, it is perfectly okay to do so. So it works right up to the point where your falling object hits the surface of the Earth. If you want to assume that you have also drilled a hole in the Earth so that the object continues to fall after it passes the Earth's surface, you will need to use a different equation to calculate its further acceleration.

But that’s the point of assuming a point mass, there is no surface to hit or hole to drill.

I think is see what your doing, your calculating the escape speeds for both r_start and r_end; then using Pythagorean theorem to find the diff.
But that has limits: thus you can’t find answers when you get close to the center.
Example: for the Sun figuring the escape speed for a spot about 1 km form the point source center. Using GM = 132 x 10⁹. You get a fall from infinity speed or an escape speed of about 360,000 km/s. Obviously we are inside Swartzchild radius. Not very useful at these limits without taking relativity into account.

But if we limit our drops to inside the solar system, just like we need to limit the relative masses so that m<<<M, we should be able to create a “straightforward” formula that can give those speeds including ‘the center’ where r=0 for the max speed befor slowing down starts, after passing the center. In orbitial terms I guess the center would become the perihelion for the "striaight line orbit" with no angular movement.


NOTE:

$- \frac{GMm}{r}$ is the gravitational potential energy
Note.as r increases, the equation becomes less negative and the potential energy increases.

That’s just a math trick of defining the MAX PE at Zero when objects are at infinity.
A good devise for figuring escape velocities using conservation. That’s not what we are doing here.

We are using the real potential energy from the start spot, back down to the center. And should be able to get speeds all the way down to & including r = 0. (Limited to cases where the speed remains well lower than c.)

It’s so standard to use the escape velocity assumptions, must be why I’m have trouble coming up with the proper way to derive this without using the familiar “negative gravitational potential energy” shortcut to get a more direct straightforward formula that dosn't need the Pythagorean theorem.

 

[Integral]

think is see what your doing, your calculating the escape speeds for both rstart and rend; then using Pythagorean theorem to find the diff.

Could you please show me where he used the Pythagoream theorem?

 

[BobG]

But if we limit our drops to inside the solar system, just like we need to limit the relative masses so that m<<<M, we should be able to create a “straightforward” formula that can give those speeds including ‘the center’ where r=0 for the max speed befor slowing down starts, after passing the center. In orbitial terms I guess the center would become the perihelion for the "striaight line orbit" with no angular movement.
NOTE:
That’s just a math trick of defining the MAX PE at Zero when objects are at infinity.
A good devise for figuring escape velocities using conservation. That’s not what we are doing here.
We are using the real potential energy from the start spot, back down to the center. And should be able to get speeds all the way down to & including r = 0. (Limited to cases where the speed remains well lower than c.)
It’s so standard to use the escape velocity assumptions, must be why I’m have trouble coming up with the proper way to derive this without using the familiar “negative gravitational potential energy” shortcut to get a more direct straightforward formula that dosn't need the Pythagorean theorem.

Janus's equation $$\sqrt{2GM \left(\frac{1}{r}-\frac{1}{r_{int}} \right)} = v$$
is pretty straight forward. It's basically Leibniz's vis-viva equation. If you substituted the semi-major axis for r_int, you could use it to find the speed of an object in an elliptical orbit, as well.

You are right that it could be used all the way down to r=0, provided all the mass really did exist in a point. Once you're below the surface, some of the mass is above you and the GM part will no longer be the same.

 

[RandallB]

Could you please show me where he used the Pythagoream theorem?

$$2GM \left(\frac{1}{r_{int}} \right)}$$
Is the square of the starting point speed.
$${2GM \left(\frac{1}{r} \right)}$$
Is the square of the ending point escape speed.
(It’s this ending point that is a problem as r nears 0.)

Combining the squares ; in this case taking the diff. since the E is being measured as a negative from infinity. And then taking a SQRT of the sum is basic Pythagoream …

You are right that it could be used all the way down to r=0, provided all the mass really did exist in a point. Once you're below the surface, some of the mass is above you and the GM part will no longer be the same.

No the point is the formula blows up when r=0

 

[SpaceTiger]

But that’s the point of assuming a point mass, there is no surface to hit or hole to drill.
I think is see what your doing, your calculating the escape speeds for both r_start and r_end; then using Pythagorean theorem to find the diff.

The problem is entirely one-dimensional, so there are no triangles involved. His method is conserving energy. When you say:

Combining the squares ; in this case taking the diff. since the E is being measured as a negative from infinity. And then taking a SQRT of the sum is basic Pythagoream

you're misunderstanding the pythagorean theorem. Taking the square root of the sum or difference of two squares does not constitute use of the pythagorean theorem unless the equivalence was derived in the context of a Euclidean space. In this case, everything he did after writing down the equation for conservation of energy was algebra and didn't require the invocation of any geometric theorems.

But that has limits: thus you can’t find answers when you get close to the center.
Example: for the Sun figuring the escape speed for a spot about 1 km form the point source center.

There are no relativistic corrections needed, even as you approach the center of the sun (or earth). Although it would appear that the equation diverges as r -> 0, you can still do conservation of energy as long as you treat the "M" in potential energy as being the mass interior to the radius of the object in question. At large distances, this is a constant (the total mass of the sun), but as you approach its center, the mass interior approaches 0.

Relativistic corrections would be necessary for the same calculation done to a black hole or neutron star.

It’s so standard to use the escape velocity assumptions, must be why I’m have trouble coming up with the proper way to derive this without using the familiar “negative gravitational potential energy” shortcut to get a more direct straightforward formula that dosn't need the Pythagorean theorem.

You can directly solve the differential equation:

$$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$$

if you like. It will give you the same answer.

 

[RandallB]

You can directly solve the differential equation:
$$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$$
if you like. It will give you the same answer.

Great
Given a start position of r = 100r_earth
A start velocity of v = 0
Defining distance d = 100r -r
What is your solution for v as a function of d ??

EXAMPLE
on the surface of the Earth I can use

V = 8 ft/sec * sqrt d

At 16 feet that gives me 32 feet per sec
Good answer
formula comes from acceleration a=32 feet per sec per sec
but it won't last long as a is getting stronger as d gets longer.

So for a complete straight forward formula
with a known starting factor of 8ft/s

V = (A*8ft/s +B) * sqrt d + C

Where A B and C are each some function of d or a constant like 1 or 0.

I just don’t know how to get to the simple formula directly calculating from d so that can be solved for d = starting radius.
So far the solutions I’ve seen strike me as incomplete integrations that havn’t accounted for taking the limit of d to it’s max of starting r. so that for d greater than starting r ; v would be decreasing from that max speed found at r=0.

 

[pervect]

You can directly solve the differential equation:
$$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$$
if you like. It will give you the same answer.

Great
Given a start position of r = 100r_earth
A start velocity of v = 0
Defining distance d = 100r -r
What is your solution for v as a function of d ??

Except for a change of variables, that would be Janus's solution. As everyone has been trying to explain.

$$\sqrt{2GM \left(\frac{1}{r}-\frac{1}{r_{int}} \right)} = v$$

r_int = 100 r_earth

BTW, suspect you really meant that
d = 100 r_earth -r

This gives

$$v = \sqrt {2}\sqrt {GM \left( \left( 100\,{\it re}-d \right) ^{-1}-{ \frac {1}{100}}\,{{\it re}}^{-1} \right) }$$

Note that a = dv(d)/dd * dd/dt = dv/dd*v = GM / (100 r_earth - d)^2

Note also that the answer, and the differential equation you started with, are both incorrect for r<r_earth, i.e for d > 99 r_earth

You will need to use the correct force law for a body inside the Earth if you really want to solve for the motion there.

 

[RandallB]

Note also that the answer, and the differential equation you started with, are both incorrect for r<r_earth, i.e for d > 99 r_earth
You will need to use the correct force law for a body inside the Earth if you really want to solve for the motion there.

Good point the differential equation given to me start with is probably the wrong way to start the solution of the problem.
As assumed before the mass is a point source. The point is to have a formula capable of solving for small r even r=0.
Compare μ for the sun is much larger than μ for earth. (μ = GM).
Thus we should expect that the Theoretical Maximum speed at crossing the center of the core point mass (without hitting it, it’s just a theoretical solution) to be much greater for the sun. But these will not solve that Theoretical Max, as they blow up at r = 0. And of course the Max Speed, obviously must occur at r= 0.

There is nothing in the formulas that demand the Mass (earth) have some fixed radius that the falling object must hit. With the mass as point source your right these formats of the solution are likely just starting from the wrong place to be able to work that solution. I just don’t know the correct place to start from.

Yes, if the mass is given a fixed diameter with uniform density or at least a changing density with a reasonable function to r , it would need adjustments to the solution I’m looking for here. This of course assumes the falling object is “transparent” to the mass (or at least someone has dug an appropriate hole.
You describe those adjustment as the “correct force law for a body inside the Earth”; is there someplace that describes doing this. Maybe some clues there.
Thanks

 

[Integral]

The point is to have a formula capable of solving for small r even r=0.

Why? This is a non physical condition, why do you expect a physical answer. For any mass there is some radius, below which you are dealing with a black hole. As stated above you need different math to deal with black holes.

A funny thing about Physics, it works best when you keep the problems physical. Your requirement is non physical therefore you can expect non physical results. Real planets have a radius, your initial problem statement was for an object between the Earth and the moon, both the Earth and moon have a radius. You have been given a solution to that problem.

 

[SpaceTiger]

You describe those adjustment as the “correct force law for a body inside the Earth”; is there someplace that describes doing this. Maybe some clues there.

The equation I gave you is correct if you substitute the mass interior:

$$M=\int_0^d 4\pi r^2\rho dr$$

This means you just need to know the density profile of the earth, or whatever object you're interested in.

 

[pervect]

If you make the assumption that the Earth has a uniform density, you get a force inside the Earth's interior that is proportional to the distance away from the center.

Of course this is an oversimplification - the Earth doesn't have a uniform density, and if you want to get really detailed answers it isn't even quite spherical.

If you do make this simplification, the potential difference below the surface follows a sqare law

V = .5*k*r^2 + C, where k = (g/R_earth) = GM/R_earth^3

Thus V(r=R_earth) - V(r=0) = .5 GM/R_earth so we have an additional (approximate!) potential drop of .5 GM / R_earth from the Earth's surface to the center

This means that the total potential from the center of the Earth to infinty is of the order of -1.5 GM/R.

This is a rather odd question to ask, though, since bodies can't actually drop through the Earth to it's center, and they are especially unlikely to drop with zero angular momentum from infinity as this would entail a non-straight tunnel through the Earth.

This potential would be the one to use if one was interested in approximating the amount of gravitational time dilation at the Earth's center relative to its surface, other than that I can't think of any physical scenario that would use it.

 

[RandallB]

Why? This is a non physical condition, why do you expect a physical answer. For any mass there is some radius, below which you are dealing with a black hole. As stated above you need different math to deal with black holes.
A funny thing about Physics, it works best when you keep the problems physical. Your requirement is non physical therefore you can expect non physical results. Real planets have a radius, your initial problem statement was for an object between the Earth and the moon, both the Earth and moon have a radius. You have been given a solution to that problem.

Assuming that the Theoretical Maximum speed I spoke of cannot be mathematically found because it doesn’t fit your idea of physical Physics is short sighted. A Ptolemaic attitude like that won’t get far. Besides we do have particles that do treat the Earth as transparent, but that’s not the point. This shouldn’t be impossible mathematics.

NO, I haven’t been given a solution. The original problem was to solve the acceleration formula as a function of distance accounting the increasing acceleration.

Using the differences between Gravitational Potential that increases as r does, is just a math trick that works for escape V’s but obviously not well here.

At a minimum it should probably be based on the formula for gravitational force that increases as r decreases. I’m no math wiz, so my problem is in finding the math; at least it’s not in seeing the concepts.

But thanks for a least trying.

 

[Integral]

For an object put into position say halfway to the moon.
Take away all orbital and any other speed – that is it’s completely stationary with respect to the earth. Of course it will immediately start to freefall directly to earth.
Acceleration is increasing so calculating the speed at different points cannot use a simple acceleration formula.
Is there a straight forward formula that gives its speed as a function of distance that accounts for the increasing acceleration?
RB

Here is your origianal problem statement. You have a good solution to this problem.

 

[SpaceTiger]

This is a rather odd question to ask, though, since bodies can't actually drop through the Earth to it's center,

Yeah, the only potential application I could think of was for a cold dark matter particle. I suspect this isn't what RandallB is interested in. :tongue2:

 

[SpaceTiger]

NO, I haven’t been given a solution. The original problem was to solve the acceleration formula as a function of distance accounting the increasing acceleration.

...

At a minimum it should probably be based on the formula for gravitational force that increases as r decreases. I’m no math wiz, so my problem is in finding the math; at least it’s not in seeing the concepts.

That's interesting, because your minsunderstandings here seem to be conceptual, not mathematical. Before we continue, let's make sure that the reasons for all of the following are clear to you:

1) There is no singularity at r=0 for the Earth or sun.
2) General relativity is not necessary unless you're doing extremely high precision calculations or working with a black hole or neutron star.
3) The Pythagorean Theorem has not been used in any of these calculations.
4) The solutions obtained by conservation of energy are equivalent to those obtained through direct integration of the differential equation.
5) Gauss' Law allows us to use the same differential equation to describe the interior of spherically symmetric objects, as long as we consider only the mass interior to the particle at each point.

If you don't understand any of these things, you should speak up so that we can explain them in more detail. Once you do understand them, it should be clear that your problem really has been solved, at least in the Newtonian limit (which is valid for all of the objects you've mentioned so far). The reason we can't give you a single explicit answer is that it depends upon the density profile, which will vary from object to object. If you specify an object, the equations can be solved for that case, but I wouldn't be inclined to spend the time generating a solution unless you're actually going to use it for something. There aren't any new physics to be learned from such a calculation.

 

[Janus]

Assuming that the Theoretical Maximum speed I spoke of cannot be mathematically found because it doesn’t fit your idea of physical Physics is short sighted. A Ptolemaic attitude like that won’t get far. Besides we do have particles that do treat the Earth as transparent, but that’s not the point. This shouldn’t be impossible mathematics.
NO, I haven’t been given a solution. The original problem was to solve the acceleration formula as a function of distance accounting the increasing acceleration.
Using the differences between Gravitational Potential that increases as r does, is just a math trick that works for escape V’s but obviously not well here.
At a minimum it should probably be based on the formula for gravitational force that increases as r decreases. I’m no math wiz, so my problem is in finding the math; at least it’s not in seeing the concepts.
But thanks for a least trying.

The concept you are failing to see is that there is no single equation that will give you the velocity of your falling object at all values of r above and below the surface of the Earth. (assuming that the mass of the Earth is distributed over its volume, but offers no resistance to the falling object.)