What is the jerk of a falling object?

[ejungkurth]

Suppose an object is suspended above the Earth, then released. It immediately accelerates to 32.2 f/s^2. There should be a jerk as the acceleration changes. Is there an experimental method to measure that jerk? Experimental as opposed to just doing the math. Thanks.

 

[berkeman]

Nope, no jerk. The acceleration is constant, but the momentum and velocity build slowly from zero.

http://en.wikipedia.org/wiki/Kinematic_equations#Equations_of_uniformly_accelerated_motion

 

[ejungkurth]

Jerk is a change in acceleration

Nope, no jerk. The acceleration is constant, but the momentum and velocity build slowly from zero.

The acceleration changes from 0 to 32.2f/s^2. Therefore there is a jerk. So I wonder if you could be more specific when you say there is no jerk.

Thanks.

 

[TVP45]

Nope, no jerk. The acceleration is constant, but the momentum and velocity build slowly from zero.

I agree that, if I tossed the same object straight up, it would go through zero velocity and would have no jerk. That object would always be accelerating at g.

But, if the object is held in place, it seems there should be a jerk as it begins to accelerate.

I'm trying to think of an experiment to test this. My jerkometer only works horizontally.

 

[berkeman]

Okay, I should have been more precise. When you remove your hand, the sum of forces goes from zero to non-zero. The acceleration does go from zero to g, but the velocity and momentum are integrals of the change in acceleration, so they change smoothly. The integral of a step function is a ramp...

 

[Mech_Engineer]

The acceleration changes from 0 to 32.2f/s^2. Therefore there is a jerk. So I wonder if you could be more specific when you say there is no jerk.

Thanks.

A jerk would be a change in acceleration with respect to time.

When you are holding the object, the acceleration and velocity is presumed to be zero, so assuming letting go of the object takes 0 seconds your acceleration instantaneously goes to -g, -32.2 ft/s^2. Therefore, the derivative (slope, a.k.a. jerk) of the step acceleration function is infinite, so the jerk is really defined as indeterminite at the point you release the object. However before and after you release the object, the jerk is 0.

I guess it should also be pointed out, that this would assume falling a small distance or neglecting air resistance. The jerk could in theory be non-zero if the object fell for a long time and you were taking into account the wind resistance on the object.

 

[TVP45]

I don't want to beat a dead horse, but...

A finite object can't be released instantaneously, so da/dt will not be infinite. Big, but not infinite. There will be a jerk.

And, because you have to start looking at it before the release, velocity and momentum will not change smoothly but abruptly at the foot of the ramp.

This is much easier to see in circular motion than in linear motion. And, it is a very small effect generally, but it's there.

 

[ejungkurth]

I'm just curious about how the effect, or lack thereof, might be measured empirically.

 

[Ulysees]

If you account for relativity's very subtle effects at low speeds, the acceleration is not constant because the inertial mass increases as the object speeds up.

So there a very small negative jerk, that only exists in the frame of the outside stationary observer, but in the frame of the object you feel no jerk if you're an observer there.

Also, the gravitational field gets stronger as the object gets a little closer to the centre of the earth. This is a positive jerk.

Which jerk is bigger? I'd go for the positive one due to the gravity gradient. Anyone care to guess the relativistic jerk, what it works out to?

 

[Ulysees]

I'm just curious about how the effect, or lack thereof, might be measured empirically.

How about getting in a lift, and someone cuts the wire holding it. You'd get a sudden jerk like in a theme park ride, and then experience what people call "zero gravity", and then experience a lot of pain as you hit the bottom.

 

[Ulysees]

I'm not saying this to you, by the way, just wanted to describe the "experience" of jerk.

 

[TVP45]

If you account for relativity's very subtle effects at low speeds, the acceleration is not constant because the inertial mass increases as the object speeds up.

So there a very small negative jerk, that only exists in the frame of the outside stationary observer, but in the frame of the object you feel no jerk if you're an observer .

Also, the gravitational field gets stronger as the object gets a little closer to the centre of the earth. This is a positive jerk.

Which jerk is bigger? I'd go for the positive one due to gravity change, an acceleration that increases by 1/6.300.000 per metre fallen. Anyone care to guess the relativistic jerk, how much it works out to?

I am absolutely convinced you can feel it. Here's the experiment I propose. Put a passenger in your car. Have the person wear their most expensive white clothes. Give them a glass of chocolate milk filled within 5 mm of the top. Find a straight steep hill. Here in Pittsburgh, we have one that goes 24 degrees. Drive carefully and smoothly onto the hill. Carefully speed up to 20 mph. Take your foot off the gas. At the instant the car reaches zero velocity, SLAM on the brakes. Look at your passenger. Run.

 

[Ulysees]

Wait a second, I take it back, what I said about the experience of jerk in a lift as the wire is cut is wrong. You don't feel any jerk at all, because you were standing on the floor, not hanging from the ceiling.

 

[Ulysees]

I am absolutely convinced you can feel it. Here's the experiment I propose. Put a passenger in your car. Have the person wear their most expensive white clothes. Give them a glass of chocolate milk filled within 5 mm of the top. Find a straight steep hill. Here in Pittsburgh, we have one that goes 24 degrees. Drive carefully and smoothly onto the hill. Carefully speed up to 20 mph. Take your foot off the gas. At the instant the car reaches zero velocity, SLAM on the brakes. Look at your passenger. Run.

 

[ejungkurth]

mech_engineer and berkeman

You guys are on track. Just so you know, I aced mechanics and vector calculus (and diffeq), but that was twenty years ago, and I'm indulging in a little speculation.

True that jerk in this situation is -g/t. But the limit as t approaches zero is negative infinity. Yes, it is indeterminate, but it is indeterminably large. So where does the yank go? Ditto for the successive derivatives of motion.

Could it be that acceleration (and the successive derivatives of motion) is (are) equally distributed over the path of the falling object, but not necessarily constant?

As I said, I have had classical mechanics drilled into me pretty well. Far be it from me to argue with Newton. So, I wonder if there has been a definitive experiment to show that there is immediate (instantaneous is not the correct term) acceleration due to gravity.

There is a seeming contradiction. But what is a seeming contradiction to me has often been explained.

Can there be a more coincidental name than Heaviside? I'm not sure that a step function explains the dilemma any better than the conjecture that a function that should approach infinity just goes to zero.

 

[ejungkurth]

PS

berkeman, you mentioned smooth acceleration. But what I am thinking is smooth motion. Smooth w.r.t. acceleration, jerk, snap, crackle, pop, ad infinitum. They said I was mad. I was able to refute them and then some. The straightjacket is actually a fashion statement.

 

[Ben Niehoff]

In the ideal limit (where an object can be "instantaneously" released), then the jerk is a delta function. A delta function (not strictly a "function", per se) is defined such that it is zero everywhere except the origin, and its definite integral over any interval containing the origin is 1.

The derivative of a delta function is an even stranger beast.

 

[Ben Niehoff]

One way you could possibly measure jerk is with an electronic device. There exist (rather simple) circuits called "differentiating amplifiers"; using an op-amp, capacitors and resistors, you can make a circuit such that the output is the derivative of the input. Then, merely tie the input voltage to something you can more easily measure, such as acceleration. Voila! A jerkometer.

Differentiating amplifiers are not used nearly as much as their cousins, the integrating amplifiers. The reason is that integrators tend to smooth out a signal, while differentiators tend to introduce noise and make the voltage blow up.

 

[ejungkurth]

Dear Mr. Niehoff,

I am intrigued by your offerings, however, I am unaware of a conjecture that relates electromagnetic forces to the forces of gravity. Can you expound?

 

[TVP45]

You guys are on track. Just so you know, I aced mechanics and vector calculus (and diffeq), but that was twenty years ago, and I'm indulging in a little speculation.

True that jerk in this situation is -g/t. But the limit as t approaches zero is negative infinity. Yes, it is indeterminate, but it is indeterminably large. So where does the yank go? Ditto for the successive derivatives of motion.

Could it be that acceleration (and the successive derivatives of motion) is (are) equally distributed over the path of the falling object, but not necessarily constant?

As I said, I have had classical mechanics drilled into me pretty well. Far be it from me to argue with Newton. So, I wonder if there has been a definitive experiment to show that there is immediate (instantaneous is not the correct term) acceleration due to gravity.

There is a seeming contradiction. But what is a seeming contradiction to me has often been explained.

Can there be a more coincidental name than Heaviside? I'm not sure that a step function explains the dilemma any better than the conjecture that a function that should approach infinity just goes to zero.

No, the jerk is only right at the beginning. You can easily see it if you graph v vs t and notice the little "bump" at the beginning of the ramp. This is a fairly well-known phenom in cam design and civil engineering; as I posted earlier it is pretty easy to see (and measure) in circular motion - tough in linear.

If you start with a point mass, then you can get immediate acceleration and that is, of course, a very different animal. But, finite objects do not have instantaneous movement. If you want, I'll figure out an experiment for you to see that.

 

[Ulysees]

> Suppose an object is suspended above the Earth, then released. It immediately accelerates

When you release the object it's like cutting a string: at the molecular level it means introducing external E/M forces between molecules that are stronger than the E/M forces holding the molecules together.

No E/M force in the universe that I know of appears instantaneously (they all tend to follow inverse-square or other continuous equations), therefore a jerk is never a true delta fuction.

An acceleration sensor would need a huge bandwidth to measure the jerk when the string is cut. I would rather attach the one plate of a capacitor to the object, and the other plate to the ceiling, and measure micro-currents with a high bandwidth oscilloscope.

 

[rszanti@compuserve.c]

The item is still accelerated (the force of gravity is still there) even though it's being held stationary in relationship to the viewer. There is no jerk because the acceleration does not change.

 

[PatPwnt]

Say you are in a spaceship accelerating at 10m/s^2 and you hold a ball out the window. You feel the weight of the ball because you are accelerating, and the ball has mass. However, when you let the ball go. Does the ball accelerate backwards? No. It simply stays at the last velocity you held it at. You are the one that feels the force of acceleration. The same is with gravity.

The ball's acceleration is constant but, you are just holding it there in place with respect to yourself. You feel the weight in your hand, therefor there is a force -> F = MA. Yea there's acceleration; it's already there.

An example of jerk is if you are in the spaceship and you increase the acceleration uniformaly, so the balls weight will get heavier when holding it. But, gravity behaves like just constant acceleration if you stand at the same radius.

An objects acceleration will increase as it falls because the force of gravity increases as you get closer to the planet, so there is jerk.

 

[ejungkurth]

I'm not sure I can accept the concept of acceleration without motion.

And it would have been more precise to factor in the change in acceleration due the decreasing inverse square of the distance. However, that isn't what has my underwear in a bunch.

The classical interpretation of the motion involved translates into saying that if there is a change in velocity in zero time, then there is no acceleration. My argument is that instead, we would be witnessing infinite acceleration.

I half expected that someone would point out that the change in acceleration (in the suspended, released body scenario) is out of phase w. r. t. the path between the centers of gravity of the respective bodies.

Note that my dilemma applies to a body at rest in free space that becomes subject to a force. I think that immediate acceleration is as counter-intuitive as immediate velocity. In other words I challenge the entire framework of classical mechanics ;)

I'm hoping that someone will (gently, please) trounce me. Otherwise I have to do math and experiments and stuff myself (sigh).

 

[ejungkurth]

Mea culpa.

Inverse square of the decreasing distance.

 

[Doc Al]

I'm not sure I can accept the concept of acceleration without motion.

Huh?

And it would have been more precise to factor in the change in acceleration due the decreasing inverse square of the distance. However, that isn't what has my underwear in a bunch.

The classical interpretation of the motion involved translates into saying that if there is a change in velocity in zero time, then there is no acceleration.

Say what? Where did you get this from? A finite change of velocity in "zero time" would of course be an infinite acceleration--but where do you think that happens? Certainly not when you drop a ball!

In elementary problems we usually neglect the time taken to remove the support force--the time to go from zero acceleration to 9.8 m/s^2. But that's just a simplification. In any case, the acceleration increases sharply (not infinitely sharply!), not the speed.

My argument is that instead, we would be witnessing infinite acceleration.

So?

I half expected that someone would point out that the change in acceleration (in the suspended, released body scenario) is out of phase w. r. t. the path between the centers of gravity of the respective bodies.

Note that my dilemma applies to a body at rest in free space that becomes subject to a force. I think that immediate acceleration is as counter-intuitive as immediate velocity. In other words I challenge the entire framework of classical mechanics ;)

Come on now. Where is this "immediate acceleration"?

 

[TVP45]

Huh?



In elementary problems we usually neglect the time taken to remove the support force--the time to go from zero acceleration to 9.8 m/s^2. But that's just a simplification. In any case, the acceleration increases sharply (not infinitely sharply!), not the speed.

Yes, you're right about the speed and I was wrong. I got preoccupied with which way the velocity vector pointed without thinking about the zero magnitude. My bad.

 

[ejungkurth]

In the ideal limit (where an object can be "instantaneously" released), then the jerk is a delta function. A delta function (not strictly a "function", per se) is defined such that it is zero everywhere except the origin, and its definite integral over any interval containing the origin is 1.

The derivative of a delta function is an even stranger beast.

I reread my previous reply and realized I sounded like an infinite jerk. I apologize.

 

[jdavel]

What if you think about it like this. Imagine a large mass suspended from a string made up of a single chain of atoms. The string is in tension with each pair of adjacent atoms held together by the electrostatic attraction between them. Then instead of cutting the string, just gradually increase the mass supported by the string until one pair of atoms pulls apart. Once the separation gets to more than about 10A, the electrostatic force is probably negligible and any jerk falls to zero. So, the jerk muct occur in that first 10A. During this interval both gravity and the decreasing electrostatic force are in play, so it's hard to say exactly how much time it would take to fall that far. But two things seem clear about the time interval: 1) it's short but 2) it's not zero.

 

[ejungkurth]

I can't figure out how to quote properly :(

>Huh?

I was replying to the poster who surmised the object was accelerating without moving.

>Say what? Where did you get this from? A finite change of velocity in "zero time" would of course be an infinite acceleration--but where do you think that happens? Certainly not when you drop a ball!

I was making an analogy to the derivative. Saying that an instant change in acceleration results in no jerk is like saying an instant change in velocity results in no acceleration.

>In elementary problems we usually neglect the time taken to remove the support force--the time to go from zero acceleration to 9.8 m/s^2. But that's just a simplification. In any case, the acceleration increases sharply (not infinitely sharply!), not the speed.

I'm not sure Newton would agree with you here. However, consider an object in free space subjected to a force.

>So?

The problem is with infinite jerk, there would be infinite yank, which is not observed. I agree with mech_engineer that the jerk is undefined, but I think it's a logical leap to equate "undefined" with "zero." However, it works pretty well in practice.

I like Mr. Niehoff's delta function construction, but in order for kinematics to hold together I think it must be a different way of saying the same thing. Then again I'm not sure how he derives the delta function from the equations of motion.


>Come on now. Where is this "immediate acceleration"?

F=ma. Another way of saying it is that an object's inertia resists a change in displacement, but not a change in velocity. Which makes sense, I think. It's the initial point that seems a bit dodgy.