How to Calculate Free Fall Time with Variable Gravity?

[simpatico]

Homework Statement

when g = constant , time = v / g
if g varies how do I get t ?

One electron is at rest at r0, 3600 km from surface of Earth (r), 10.000 km from center.
g(r) = 9.8 m/s^2 , g(r0) = 4, Δ PE = 2.25 x 10^7

1) If it fell in a vacuum, after how many seconds would touchdown?

Homework Equations

(a) v = sqrt( 2x PE), (b) g= PE/ h

The Attempt at a Solution

from (a) => v = sqrt ((2 x2.25) 4.5 x 10^7) = 6,708 m/s
from (b) => average g = 2.25 x 10^7 / 3,600,000 = 6.25 m/s^2, t= 6708/6.25= 1273 s

Homework Statement

2) ceteris paribus, if we change gravity with elecrostatic attraction, then drop of Potential becomes 0.0009305 eV,and final v seems to change from 6,7 to 18,1 km/ s.WHY?

Homework Equations

The Attempt at a Solution

no idea

 

[gneill]

Can you write an expression that would give the speed of the falling particle for any given radial distance? (HINT: conservation of energy)

The speed of the particle is -dr/dt at radius r. (dr/dt is negative because the radius is decreasing).

Looks like an integration is required...

 

[elegysix]

saying g varies, you imply that acceleration varies... look at Newton's second law,

F=ma, where a is now function of R, such as a=G/R^2

****hint, setting up the DE****

dV/dt=-ma(R)

and a change of variables:

(dV/dt)=(dV/dR)(dR/dt) = V(dV/dR)

gives V(dV/dR)=-mG/R^2

 

[Studiot]

Hello simpatico, welcome to Physics Forums.

You have not said what your maths level is , but I am guessing this is a high school question?


Have you covered the equations of motion for constant acceleration?

When the acceleration is not constant each of the quantities v (edit =instantaneous velocity); s (=distance); a (=acceleration) will vary with t (=time). A graph may be drawn connecting any two of these four quantities. If this can be done the other quantities can be deduced from this graph.

Alternatively the result may be calculated by integration as already indicated, but this approach is not taught in the UK at high school level.

Averaging will not work as it did to obtain the constant acceleration formulae.

Over to you.

 

[simpatico]

Thanks Studiot,
The problem is high-school,
but a Prof in Mechanics in UK, in another Forum ,solved the problem using average g finding 1021.5 sec.
I was not convinced and gave it a try, and found 1273, WHICH is right or better?

Just what I need, if you haven't the time to work out the exact result, is to know

Using the procedure of average g., the way I calculated,you say : it won't work, ok, but...

can I obtain a good approximation for TIME? ( say: 0,0 1)
Thanks

P.S. Can I legitimately expect that someone will give me the right answer?

 

[Studiot]

I have already suggested drawing a graph, if you want to avoid integration.
Note I made a slight correction to my earlier post in the velocity term.

The area under an acceleration - distance graph equals yields the velocity.

You can draw this since you have been given the acceleration as a function of distance in post#3

The proof uses integration, but you do not need to do so

$$\int {ads = \int {v\frac{{dv}}{{ds}}} } ds = \int {vdv = \frac{1}{2}} {v^2}$$

So the area under the accel-dist graph from zero to any distance s' gives $$\frac{1}{2}{v^2}$$ where v is the velocity at s'.

 

[simpatico]

Can someone PLEASE answer this simple question:

If (instead of integrating variable g)

I use average g (ΔPE= KE/ h) and the formulas I normally use when g = constant

what approximation do I get for TIME (say 0.03)?

am I asking too much?

 

[Studiot]

Averaging will not work as it did to obtain the constant acceleration formulae.

This was answered in post#4

 

[simpatico]

This was answered in post#4

I got so far, my question ( in post #5) is very simple too, HOW BAD doesn't work ?
If I need a rough solution, is it always OK?

 

[simpatico]

Hi helpers and mentors
I've done my homework,
(is this formula correct?
t = 1/ (sqrt(GM/r0^3) * (π/ 2 + sqrt (r/ (r0 * (1-(r/ r0))- arcsin(sqrt(r/ r0)) )


Could anyone, please tell me if it is right?

P.S.
(In case you are reluctant to help because you think I am a teenager cheating with his homework), I'm pushing 70 !

 

[gneill]

1258 secs is the accurate result.

 

[ideasrule]

Using your method, you should get 1073 seconds. 6708/6.25 is 1073, not 1273.

I can confirm that the answer should be 1258 s. (I actually got 1257 s, but the difference is negligible.)

 

[simpatico]

6708/6.25 is 1073, not 1273.

yes, thanks, I soon realized the misprint, but the system didn't let me edit it
So 1073/ 1257(8) = 0.85 %: bad enough, just to get a rough idea !

Now, folks, as you have become so cooperative since you discovered I'm an old codger, could you please oblige me further?o

1) If I approximate by average, shouldn't I get a Higher figure? is there another mistake hidden somewhere?
2)(second question in thread)
why in a drop of PE in an electric (not gravitational) field with the same data I get
a different final velocity 18.1 m/s

Thanks

 

[gneill]

The average that you're using is a spatial one, based upon two values separated by some distance; it's not a time-averaged acceleration. So, for example, the time taken to cross the second half of the trajectory is significantly less than the time to cover the initial half. When you apply the average (but constant!) acceleration to the particle, time to cross the upper section of the trajectory is substantially reduced, and in the lower regions where the 'true' acceleration should really be higher, the particle spends much less time crossing, so its influence isn't as great. So the total time is less for the average acceleration.

You can confirm this, in a way, by calculating the final velocity of the particle for the two cases. Use the change in potential energy to find the change in kinetic energy. In one case you use the 'true' gravitational potential change from Newton's law (GMm/r), and in the second (average) case, assuming a uniform gravitational field, ΔE = m*g*Δh.

For your second question, I think you'll have to be more specific about the details.

 

[simpatico]

Thanks , you have been really GREAT help.

The details of the second question are the same as first:
just substitute Gravity with electricity.
But never mind , probably someone else will take care of that!

 

[ideasrule]

Now, folks, as you have become so cooperative since you discovered I'm an old codger, could you please oblige me further?o

We (or at least, I) treat everybody the same way, regardless of age. I think it was reasonable to assume that this was a homework question, because you posted in the "Homework & Coursework Questions" forum, whereas general discussion of physics should go here: https://www.physicsforums.com/forumdisplay.php?f=111.

The details of the second question are the same as first:
just substitute Gravity with electricity.

What charge are you assuming for the Earth, and why that value? Whatever value you're using, why do you think 18.1 km/s is not a reasonable final speed?

 

[simpatico]

thanks ideasrule,
(wasn't it right to use the homework section, since I needed an expert to check my calcs?)

now...

just suppose at center of Earth you have a positve charge with attractive force equal to GM (whatever it is)

an electron is in free fall from 3600km(...same data...ceteris paribus means: all other data being equal) Δ PE = 0.00093 eV

shouldn't final velocity at 6.4x 10^6 (ground) be the same namely 6.7 km/ s?
thanks again

 

[gneill]

The charge on the electron is not the same as its mass, so you'd need to adjust the "Earth" charge to take this into account, perhaps by setting the charge so that the force on the electron at its initial separation is the same as the force that gravity had produced (equivalently, so that the PE's are the same).

If this is done, then since the final velocity should depend only on the change in PE, they should be the same.

 

[simpatico]

The charge on the electron is not the same as its mass.

I can't see the point. (I chose one electron on purpose as it is
the elementary unit of charge: the unit Coulomb is just 6,24x10^18 charge units and
the unit of mass,: 1 electron or 1 kg receive the same acc.)

What I thought matters is only attractive force, (that's why I left it unspecified ,Im not sure
what is the exact charge that would produce such drop of potential)
You calc it and substitute.Where do I go wrong?

(P.S. my calcs gave 11.5 million charges = 0.0055 esu. but I think it makes no difference)

 

[gneill]

I can't see the point. (I chose one electron on purpose as it is
the elementary unit of charge: the unit Coulomb is just 6,22x10^18 charge units and
the unit of mass,: 1 electron or 1 kg receive the same acc.)

That's fine. Suppose you substitute a proton for the electron (and change the sign of the attracting charge). The magnitude of the charge on the particle is the same as before, so the attractive force is identical to the electron case. However, the particle mass is 1836 times greater, and so the acceleration a = F/m, will be 1836 times less. This will affect the velocity profile (KE).

What I thought matters is only attractive force, (that's why I left it unspecified ,Im not sure
what is the exact charge that would produce such drop of potential)
You calc it and substitute.Where do I go wrong?

Perhaps you could calculate and post a suggested value for the charge? [EDIT: Never mind, I see you added it to your previous post -- 11.5 million e⁺. Looks a bit small to me, I would have thought closer to 1.5 Billion e⁺ charges, or about 0.252 C.]

 

[simpatico]

That's fine. Suppose you substitute a proton for the electron post --
2) 11.5 million e⁺. Looks a bit small to me, I would have thought closer to 1.5 Billion e⁺ charges, or about 0.252 C.]

1) That's exactly why I didn't choose a proton, because the comparison holds with an electron.My field is theoretical physics. I am only interested in abstract problems and incoherences

2) ooops, that's why I'm reluctant to calc, I forgot that esu is in cm, so it is 15 billion
but, I repeat, it is no relevant ,
when you calc final v, it is different. Try to explain,
but be prepared for a shock: there is no explanation. it's just like that!

3) then comes the most interesting part of the problem
times of free fall MUST be different from 1258 sec, because

a) you cannot integrate any more on space, but on time
b) acceleration is not any more increasing but decreasing

Well, I hope you are not fed up, rather... that you are intrigued and are willing to take this problem to completion, since no one else has shown up.

 

[gneill]

A slight correction to my previous post (it's too late to edit it): the required charge to represent the 'Earth' for the falling electron would be 0.252 μC. (I forgot to paste in the μ).

For an electron located at radius r_o = 10,000km from that charge, the potential energy is the same as that of the electron mass in the Earth's gravitational field at that radius, namely about 3.631 x 10^-23J. At radius r_f = 6400km the situation is similar, with both potential energies amounting to 5.673 x 10^-23J.

The difference in potential energies at the two radii yields the kinetic energy of the electron at impact. It is the same in both cases. So I'm not sure what your issue is

 

[simpatico]

my issue is the same:
you found the right charge which produces the same drop of potential energy.Right.
Now find out the velocity at ground is it 6.7 km/sec . ?
I think you already replied that it is not.
right.
Now, as I said, if we want to find out TIME,
we must integrate on time because not acceleration but KE is growing
and acceleration, on the contrary, from zero at r0 (rest), jumps to a maximum after one second and then decreases
Can you find time of free fall in an electric field?

 

[gneill]

my issue is the same:
you found the right charge which produces the same drop of potential energy.Right.
Now find out the velocity at ground is it 6.7 km/sec . ?

I think you already replied that it is not.

No, it's the same as in the gravitational case if the charge is chosen so as to duplicate the potential energies at start and finish. This must be so because the change in PE is the same in each case, hence the change in KE is also the same.

right.
Now, as I said, if we want to find out TIME,
we must integrate on time because not acceleration but KE is growing

Both are growing as the particle falls; the force is inversely proportional to the square of the separation.

and acceleration, on the contrary, from zero at r0 (rest), jumps to a maximum after one second and then decreases

You will have to explain your reasoning here. What makes you think that the electron will jump to a maximum acceleration after one second? Why should it decrease thereafter?

Can you find time of free fall in an electri field?

Use the same methodology that you did in the case of gravity earlier in the thread.

 

[simpatico]

1)if the charge is chosen so as to duplicate the potential energies at start and finish.

2)You will have to explain your reasoning here. .

1) if you duplicate charge we do not have same conditions, but let's forget.that is settled.!

2) It's me confused, now, as you keep using Coulombs.I knew that in electrostatics I must use cgs.Never mind
cgs
So I need 5523 esu ( I hope this time I got it right)
the electron is at rest, at 10^ 9 cm up high), after 1 second it as a speed of 65 cm/ s.
acc = 65 (65 -0)
after 2 second speed = 92 cm/ s
acc = 27 (92 - 65) and so on , up to speed (18,1 km or) 670000 cm/s, acc 0.0011
in this case KE, speed increases slowly with time not ACCELERATION

(never mind if the figures are not exact , what count is principles, procedure)

Hope I made myself clear.

as to integration, you did it !.I haven't a clue

 

[gneill]

I am confused now, as you keep using Coulombs.I knew that in electrostatics I must use cgs

There is no particular reason to use cgs units. The physics remains the same regardless of the unit system. It just happens that the mks unit system is the one most commonly used here.

So I need 5523 esu ( I hope this time I got it right)

I make it 756 esu. An esu is equivalent to 3.3356 x 10^-10 C.

the electron is at rest, at 10^ 9 cm up high), after 1 second it as a speed of 65 cm/ s.
acc = 65
after 2 second speed = 92 cm/ s
acc = 27
in this case KE, speed increases slowly with time not ACCELERATION

(never mind if the figures are not exact , what count is principles, procedure)

Hope I made myself clear.
as to integration, you did it.I haven't a clue

The speed will increase in precisely the same manner as it did in the case of gravitation, as the force, change in PE, and acceleration, will be the same over the trajectory.

Integration of the equation of motion is required in order to get precise results.

 

[simpatico]

The speed will increase in precisely the same manner as it did in the case of gravitation, as the force, change in PE, and acceleration, will be the same over the trajectory.

.

Please state what is speed afterone and two seconds

 

[gneill]

Please state what is speed after one and two seconds

v(1 second) = 3.986 m/s

v(2 seconds) = 7.972 m/s

Does that help?

 

[simpatico]

v(1 second) = 3.986 m/s

v(2 seconds) = 7.972 m/s

Does that help?

well if it is so, then time will be 1258 secs.

Thanks, Gneill, for your help and patience!