Three quick questions about falling

[Talakeal]

A few quick questions about falling if I may:

First, someone on one of the RPG boards mentioned that although falling speed continues to increase, the actual force of the impact is linear with the distance fallen. Is this correct?

Second, I know that objects have different terminal velocities based on their size and shape. Does the amount of time it takes them to reach terminal velocity also vary? If so how"?

Three, if I am in free fall and I suddenly pick up a massive object with no downward momentum (say by grabbing a tossed bowling ball while it is at the apex of its movement) will the added mass decrease my falling speed?

Thanks!

[gomipile]

A few quick questions about falling if I may:

First, someone on one of the RPG boards mentioned that although falling speed continues to increase, the actual force of the impact is linear with the distance fallen. Is this correct?

Ignoring air resistance and also ignoring the reduction of gravity due to height, the kinetic energy of a falling object is linearly proportional to the distance it has fallen if it started at rest. Impacts are complex interactions that can't be characterized by something as simple as a single number for force, so we don't generally refer to "impact force."


Second, I know that objects have different terminal velocities based on their size and shape. Does the amount of time it takes them to reach terminal velocity also vary? If so how"?

Yes. A feather reaches terminal velocity nearly instantly when released, while a heavy weight takes quite some time. I don't know the exact relationship off the top of my head.


Three, if I am in free fall and I suddenly pick up a massive object with no downward momentum (say by grabbing a tossed bowling ball while it is at the apex of its movement) will the added mass decrease my falling speed?

Thanks!
No. In that example, the bowling ball adds mass without adding much drag, so it should increase the falling speed of the combined body (you+bowling ball.)

[Douglas]

No. In that example, the bowling ball adds mass without adding much drag, so it should increase the falling speed of the combined body (you+bowling ball.)
In the eventual terminal velocity scenario, yes the bowling ball would probably speed him up. In the more immediate "he just grabbed it this instant" scenario, it would slow him down because his momentum is unchanged but spread over more mass. This would be felt as a sudden jerk starting at the hand(s) that grabbed the ball and traveling through the body from there.

[Citizen Nij]

A few quick questions about falling if I may:

First, someone on one of the RPG boards mentioned that although falling speed continues to increase, the actual force of the impact is linear with the distance fallen. Is this correct?
What 'e said.


Second, I know that objects have different terminal velocities based on their size and shape. Does the amount of time it takes them to reach terminal velocity also vary? If so how"?
Acceleration due to gravity is assumably constant in velocity. Acceleration due to drag (which is in opposition to the direction of motion, in this case opposite to gravity, and based on the factors you mention) is a quadratic in velocity.
The point at which these cancel to zero is terminal velocity, and can be calculated if you know the appropriate values and equations. Can't derive a useful relation between vt and Cd just now, though.


Three, if I am in free fall and I suddenly pick up a massive object with no downward momentum (say by grabbing a tossed bowling ball while it is at the apex of its movement) will the added mass decrease my falling speed?
If you can somehow catch and hold the ball stationary to yourself, so that it is forced to accelerate faster than gravity alone would cause, then yes. Net momentum must be the same - an increase in mass must result in a proportionate decrease of velocity.
However the ball will have some reasonable inertia, which means the ball will resist being accelerated quickly, which will lessen the effect of any change in your own velocity. The result will be a noticeable difference between the calculated velocity after you grab the ball, and the actual velocity when doing so. You'll still slow down a little, but not by as much, because the ball will catch up due to acceleration by gravity.

[factotum]

The result will be a noticeable difference between the calculated velocity after you grab the ball, and the actual velocity when doing so. You'll still slow down a little, but not by as much, because the ball will catch up due to acceleration by gravity.

That's assuming the human body is a completely rigid point mass, though, which it clearly isn't. What would probably happen is that a lot of the momentum transfer would go into making the falling Talakeal rotate rather than directly slowing him down.

[Killer Angel]

That's assuming the human body is a completely rigid point mass, though, which it clearly isn't.

Indeed. if you combine a human that actively tries to modify it's "flying attitude" and the air resistance, you definitely can obtain significant variations of the falling speed.

[Tirunedeth]

Yes. A feather reaches terminal velocity nearly instantly when released, while a heavy weight takes quite some time. I don't know the exact relationship off the top of my head.

For quadratic drag, linear drag, or both (probably both, based on dimensional considerations), the terminal velocity vt and the time constant τ are related by gτ = vt, where g is the gravitational field strength (usually about 9.8 m/s2). What precisely the time constant means varies between quadratic and linear drag, but you can estimate that an object will reach terminal velocity after a few (probably five or so) time constants.


That's assuming the human body is a completely rigid point mass, though, which it clearly isn't. What would probably happen is that a lot of the momentum transfer would go into making the falling Talakeal rotate rather than directly slowing him down.

Angular momentum and linear momentum are separate conserved quantities, so the fact that a person is an extended object wouldn't have any effect on changes to linear velocity from picking up the bowling ball. All it would do is add an additional rotational component.

[Kato]

1) To be honest, I'm not entirely sure what the impact force depends on. My guess would be velocity, an since you may get roughly faster linear in time, your height vs velocity in an ideal (vacuum) case is quadratic, isn't it? (v=a*t and s=a/2 t², so s~v². Of course drag and such could likely lead to a more linear relationship)

2) As was said, yes, it depends on the drag/terminal velocity. If you have basically no air resistance, you will take long/forever to reach terminal velocity. If your drag is huge, i.e. you have a parachute you'll reach terminal velocity very fast and it will be rather small.

3) At the apex of it's movement suggests it has no upward momentum, so it will just make you heavier. Which, from all I've learned in physics, assuming your air resistance stays the same, means you won't slow down or speed up. But you'll hit the ground harder/with more energy because more mass.
If we're not talking about at its apex, consder the following: imagine someone shoots a rocket at you while you fall and you hold on to the rocket. Would you slow down? Okay, in all likelyhood, it would rip your arms off, if you managed it, but in an ideal case, yeah, something with a momentum upwards added to your momentum downwards ideally slows you down or reverses your momentum.

[Talakeal]

So as to the third question:

Basically, my understanding is that momentum equals mass x velocity. As we have conservation of energy going for us, I would imagine the momentum wouldn't change here, but the mass of the falling body is effectively increasing, thus we would need to reduce velocity accordingly. Is my reasoning correct? If not why?



Also, last night in a different discussion someone claimed that if an object is no longer resting on the Earth the Earth will rotate out from under it and hundreds of miles an hour.
This seems plausible at first glance, as I know that it is only friction which keeps something from sliding off a rotating object; but objects which are falling, flying, launched into the air, or hovering clearly do not go shooting off in the opposite direction to the Earth's rotation.
Could someone explain to me how this works or does not work?

[Tirunedeth]

So as to the third question:

Basically, my understanding is that momentum equals mass x velocity. As we have conservation of energy going for us, I would imagine the momentum wouldn't change here, but the mass of the falling body is effectively increasing, thus we would need to reduce velocity accordingly. Is my reasoning correct? If not why?

Your conclusion is correct, but your reasoning isn't quite. Momentum and energy are separate conserved quantities, so conserving one doesn't necessarily conserve the other. Otherwise, you're absolutely correct. If you have a mass M and are falling at speed V, and grab an object with mass m falling at speed v, your speed afterwards will be
V' = (MV + mv)/(M + m)
Where the numerator is the initial momentum and the bottom is the final mass. Note that this won't conserve energy (or, at least, kinetic energy; really, the energy gets converted into heat, sound, etc), since
Ei = (MV2 + mv2)/2
Ef = (M + m)V'2/2 = (MV + mv)2/(2*(M + m))
which are pretty clearly not equal.


Also, last night in a different discussion someone claimed that if an object is no longer resting on the Earth the Earth will rotate out from under it and hundreds of miles an hour.
This seems plausible at first glance, as I know that it is only friction which keeps something from sliding off a rotating object; but objects which are falling, flying, launched into the air, or hovering clearly do not go shooting off in the opposite direction to the Earth's rotation.
Could someone explain to me how this works or does not work?
Sort of. If something is at rest on the surface of the Earth, gravity (not friction) is keeping it on the surface. Without some force acting on it, the object would move in a straight line. That straight line will carry it away from the Earth, which is where centrifugal force comes from (and centrifugal force is a real thing; it just isn't a force). Of course, if you change your distance from the axis of rotation, you need a different speed to maintain the same angular speed, which is where the Coriolis effect comes from. Those two fictitious forces let you describe motion in a rotating reference frame.

I feel like that may not be entirely clear, so let me try restating it. The Earth rotates underneath objects above it, but those objects don't lose the speed they had when they were on Earth's surface just because they aren't in contact any more. However, barring other forces the object will have a different angular speed as it moves further from the axis of rotation, which creates fictitious forces in a rotating reference frame.


1) To be honest, I'm not entirely sure what the impact force depends on. My guess would be velocity, an since you may get roughly faster linear in time, your height vs velocity in an ideal (vacuum) case is quadratic, isn't it? (v=a*t and s=a/2 t², so s~v². Of course drag and such could likely lead to a more linear relationship)

At least if you aren't destroying the thing you're hitting, it seems reasonable to approximate an impact as a Hooke's law spring. In that case, you wind up getting a maximum displacement (and thus maximum force) proportional to velocity.

Anyways, now that I'm not working on completely insufficient sleep and in the middle of doing a bunch of grading, I actually feel like looking up the properties of vertical motion with quadratic drag. The equation for velocity as a function of time is
v(t) = vter*tanh(t/τ)
where τ = vter/i]/[i]g and vter is terminal velocity. This actually levels out very quickly, so within about 2τ to 3τ the velocity is nearly equal to terminal velocity.

[The Evil DM]

Tirundeth has covered energy and momentum exchanges quite adequately. The only issue I see in his comparison of kinetic energy and momentum is omitting that momentum is a vector quantity, while kinetic energy is scalar. Assuming the analysis is limited to completely 1-d the equations are correct. But a 1d assumption is totally unrealistic due to the human body being non-rigid.

I can shed some more light on Impact forces.

The best way to analyze total forces in impact is through use of Impulse Momentum Theorem. The challenge with Impulse Momentum Theorem is for any suitably realistic application that can yield some results worth generalizing it requires at a minimum integral calculus for analysis.

Impulse is the change in momentum. In general I = minitial*vinitial - mfinal*vfinal

Impulse also equals the integral of Force over the Time of impulse.

so Integral(F(t)dt) = mfinal*vfinal - minitial*vinitial

In the case of a falling impact, final velocity is zero relative to the surface of impact.

Integral(F(t)dt) = - minitial*vinitial

Solving that integral requires some assumptions about the function F(t) and some good information about impact mechanics can be found here on wiki.

http://en.wikipedia.org/wiki/Impulse_(physics)

[Douglas]

Also, last night in a different discussion someone claimed that if an object is no longer resting on the Earth the Earth will rotate out from under it and hundreds of miles an hour.
This seems plausible at first glance, as I know that it is only friction which keeps something from sliding off a rotating object; but objects which are falling, flying, launched into the air, or hovering clearly do not go shooting off in the opposite direction to the Earth's rotation.
Could someone explain to me how this works or does not work?
Inertia. An object resting on the Earth's surface is moving at the same speed the Earth's surface is. An object in motion tends to stay in motion, as per Newton's First Law. Lift the object off Earth's surface, and you get an object that is above Earth's surface but still drifting along at about the same speed.

To get the "Earth rotate's out from under it" effect, you would need to actively accelerate the object in the direction opposite Earth's rotation, and this acceleration would have to be quite high.

[BannedInSchool]

Another way of looking at it is if you were sliding on a frozen lake on a sled and someone held out a bowling ball for you to run into. You'll go slower and the ball will go faster after the impact. If the "ball" were instead a whole person you ran into it's more obvious that's what would happen. If you happened to be sledding down a hill it still works the same even with gravity supplying acceleration; that's just some velocity added in over time on top of what happens with the collision.

[Sith_Happens]

Impacts are complex interactions that can't be characterized by something as simple as a single number for force, so we don't generally refer to "impact force."

A bit more specifically, how much force you hit the ground with depends just as much on how long it takes the ground to stop you as it does on how fast you were going. A bit more specifically than that, .


Your conclusion is correct, but your reasoning isn't quite. Momentum and energy are separate conserved quantities, so conserving one doesn't necessarily conserve the other. Otherwise, you're absolutely correct. If you have a mass [i]M and are falling at speed V, and grab an object with mass m falling at speed v, your speed afterwards will be
V' = (MV + mv)/(M + m)
Where the numerator is the initial momentum and the bottom is the final mass. Note that this won't conserve energy (or, at least, kinetic energy; really, the energy gets converted into heat, sound, etc), since
Ei = (MV2 + mv2)/2
Ef = (M + m)V'2/2 = (MV + mv)2/(2*(M + m))
which are pretty clearly not equal.

As a side note, this is the difference between an elastic and inelastic collision: An elastic collision is defined as any collision in which total kinetic energy is conserved.


Inertia. An object resting on the Earth's surface is moving at the same speed the Earth's surface is. An object in motion tends to stay in motion, as per Newton's First Law. Lift the object off Earth's surface, and you get an object that is above Earth's surface but still drifting along at about the same speed.

To get the "Earth rotate's out from under it" effect, you would need to actively accelerate the object in the direction opposite Earth's rotation, and this acceleration would have to be quite high.

Slightly shorter version: As long as gravity is the only thing accelerating you your horizontal position relative to the Earth's surface will remain unchanged.

[Binks]

This seems plausible at first glance, as I know that it is only friction which keeps something from sliding off a rotating object; but objects which are falling, flying, launched into the air, or hovering clearly do not go shooting off in the opposite direction to the Earth's rotation.
Could someone explain to me how this works or does not work?
You're not keeping up with the Earth as it moves because of friction with the ground, you're keeping up with the Earth as it moves because you're moving at the same velocity. You're moving at the same velocity because you've always* been moving at the same velocity. Your mom was moving at the same velocity when she had you, and her mom, and so on. We don't notice the fact that we're moving at 1000+ mph because everything we deal with on a day-to-day basis is doing the same thing. Velocity != Acceleration.

All those examples (flying, falling, launched) are things that, at one point or another, were sitting stationary relative to Earth's surface, so they had the same velocity. None of them were subjected to anything that would remove that velocity, so they still have it, even when they're not touching Earth's surface.

* Technically your velocity is tangent to Earth's surface and is constantly changing its vector, with its magnitude remaining relatively constant (changing only if you go north or south). This change is an acceleration but it's countered by gravity (it's <1% the acceleration of gravity from the Earth in the opposite direction, so you don't even notice it).

[Talakeal]

You're not keeping up with the Earth as it moves because of friction with the ground, you're keeping up with the Earth as it moves because you're moving at the same velocity. You're moving at the same velocity because you've always* been moving at the same velocity. Your mom was moving at the same velocity when she had you, and her mom, and so on. We don't notice the fact that we're moving at 1000+ mph because everything we deal with on a day-to-day basis is doing the same thing. Velocity != Acceleration.

All those examples (flying, falling, launched) are things that, at one point or another, were sitting stationary relative to Earth's surface, so they had the same velocity. None of them were subjected to anything that would remove that velocity, so they still have it, even when they're not touching Earth's surface.

* Technically your velocity is tangent to Earth's surface and is constantly changing its vector, with its magnitude remaining relatively constant (changing only if you go north or south). This change is an acceleration but it's countered by gravity (it's <1% the acceleration of gravity from the Earth in the opposite direction, so you don't even notice it).

I was under the impression that velocity went in a straight line. AFAIK the Earth is rotating, we should simply fly forward and eventually out into space if gravity and friction suddenly ceased to apply, just like an object placed on a rotating record would slide off if not for friction.

[Tyndmyr]

I was under the impression that velocity went in a straight line. AFAIK the Earth is rotating, we should simply fly forward and eventually out into space if gravity and friction suddenly ceased to apply, just like an object placed on a rotating record would slide off if not for friction.

Nah, gravity holds us on earth, and thus we spin with it. In our frame of reference, we're not really moving relative to the earth, but from an external frame of reference, the whole glob is spinning. Us, air, earth, all of it.

It's no different than being in a car at a constant speed, you feel mostly stationary unless quick changes happen in direction or speed.

And if that happened for the earth, we would indeed be quite bothered by it.

[Icewraith]

Note that trying to grab a bowling ball with "no downward momentum" while falling at a certain speed is the about the same as being stationary and being hit with a bowling ball moving at the same net speed (the gravity vectors are different). You'd want the bowling ball to be moving at approximately the same speed as you before trying to interact with it.

Otherwise, the resulting collision will probably kill the falling person in an extremely messy way. Since the resulting bits will have significantly reduced cross sections and significantly reduced mass, drag is an exponential function of cross-section and most of the remaining bits will fall faster. The bowling ball will gain some downward momentum from the collision. If it was already unaccountably hanging in the air and ignoring gravity while remaining a bowling ball, in theory it will continue moving with the same velocity until air resistance reduces its velocity to zero or it hits the ground. If the bowling ball was instead at zero velocity because it was at the apex of a ballistic trajectory when the collision occurred (perhaps it is a cannonball instead?), it will gain some extra downward momentum from the collision and will continue on a slightly altered ballistic trajectory. The initial conditions of the collision may also result in large moments and impart opposite angular velocities to the bowling ball and debris cloud, respectively. If the bowling ball is spinning its trajectory may deviate further from its pre-collision ballistic path.

Note that any person bits that are flimsy enough may display some feather-like aerodynamic qualities, and so their paths to the ground will be slower and more varied than a simple gravity + drag calculation would indicate.

[Douglas]

I was under the impression that velocity went in a straight line. AFAIK the Earth is rotating, we should simply fly forward and eventually out into space if gravity and friction suddenly ceased to apply, just like an object placed on a rotating record would slide off if not for friction.
Friction is irrelevant to this, and gravity just keeps us on the ground. For the record analogy, consider a record with no friction that rotates so slowly it takes a full 24 hours to complete a single full turn.

If gravity magically stopped one day (but only for people), the immediate result would simply be that everyone would start gently floating in the air like astronauts in free fall, staying in near perfect synchronization with their surroundings. The rotation would manifest as a very gentle drift directly upwards, barely noticeable in short time spans, with no noticeable horizontal drift at all without extremely sensitive scientific instruments. Someone inside a building would occasionally have to push off the ceiling, but would have substantially long intervals between. He would not notice any horizontal effect. Someone outside, ignoring wind, would drift upward 1 foot in... about 4 seconds. It would take 4 minutes for the rotation to bring Earth's surface 1 kilometer away from him, but even that high up he'd still be above essentially the same spot of ground, with no meaningful horizontal drift.

[Sith_Happens]

I was under the impression that velocity went in a straight line. AFAIK the Earth is rotating, we should simply fly forward and eventually out into space if gravity and friction suddenly ceased to apply, just like an object placed on a rotating record would slide off if not for friction.

By that logic it would be impossible for anything to orbit anything else.

[Icewraith]

By that logic it would be impossible for anything to orbit anything else.

Orbiting things would probably be pretty difficult if gravity indeed suddenly turned off.

[Sith_Happens]

Orbiting things would probably be pretty difficult if gravity indeed suddenly turned off.

Whoops, missed the part about gravity, though he'd just said "if friction turned off.":smallredface:

[Kato]

If gravity magically stopped one day (but only for people), the immediate result would simply be that everyone would start gently floating in the air like astronauts in free fall, staying in near perfect synchronization with their surroundings. The rotation would manifest as a very gentle drift directly upwards, barely noticeable in short time spans, with no noticeable horizontal drift at all without extremely sensitive scientific instruments. Someone inside a building would occasionally have to push off the ceiling, but would have substantially long intervals between. He would not notice any horizontal effect. Someone outside, ignoring wind, would drift upward 1 foot in... about 4 seconds. It would take 4 minutes for the rotation to bring Earth's surface 1 kilometer away from him, but even that high up he'd still be above essentially the same spot of ground, with no meaningful horizontal drift.

Hm, I was about to argue but it seems the numbers ARE right. Rather surprising, to me. Of course, this is only at the equator, depending on your latitude it would vary strongly, and with time the number would grow fast. On the other hand, depending on what you'r standing on, there is a good chance you wouldn't loft off the ground because the ground would be flung out together with you, without gravity holding it back. Though, possible earth yould instantly expand and you'd fly far more straight upwards? I think we need a geologist for this.

[Knaight]

Hm, I was about to argue but it seems the numbers ARE right. Rather surprising, to me. Of course, this is only at the equator, depending on your latitude it would vary strongly, and with time the number would grow fast. On the other hand, depending on what you'r standing on, there is a good chance you wouldn't loft off the ground because the ground would be flung out together with you, without gravity holding it back. Though, possible earth yould instantly expand and you'd fly far more straight upwards? I think we need a geologist for this.

As regards the hypothetical instant expansion of the earth if the matter in it all suddenly stopped exerting a gravitational force - probably not. Even the atmosphere wouldn't vanish nearly instantly, instead gradually diffusing outwards. The actual velocities of individual atmospheric particles are generally in the few hundreds of meters per second range (though this obviously varies by chemical species, local temperature, individual particle, etc. Most end up under 1000 m/s, though the average for hydrogen is way above that). So, the atmosphere will gradually disappear, and the pressure drop will generally take liquids with it. Rocks and similar things will remain solid at zero pressure and typical temperatures. The chemical bonds and intermolecular forces continue to hold the ground together decently.

With that said, over a longer period the gravity absolutely is key, and some sort of slower expansion is to be expected.

[Max™]

I'm just picturing people trying these experiments, someone jumping off a platform with someone below holding a bowling ball and letting it go just as the person is about to go past, WHACK!

Then the person who let the bowling ball go turns to the camera and smiles, a caption pops up: "Scientists are jerks.", fade to black.