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Thread: A problem with F=ma

  1. #1 A problem with F=ma 
    Forum Bachelors Degree Waveman28's Avatar
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    F=ma cannot be correct. Consider projectile motion, we throw a 1kg ball in the air and at some point it will be totally stationary in the air with no velocity. At this point, F=ma so the force is 9.8N. How can this be if the ball is not even moving? This happened the other day when I was trying to calculate the force on a moving object. What do I do in this situation? This may seem like a very elementary question but I know of no other formulas for this situation.


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  3. #2 Re: A problem with F=ma 
    Forum Ph.D. Leszek Luchowski's Avatar
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    Quote Originally Posted by Waveman28
    At this point, F=ma so the force is 9.8N. How can this be if the ball is not even moving?
    This is not a trivial question at all. This kind of questions led great minds to discuss the nature of motion, speed and time for millennia - in fact, from antiquity (see Zeno's paradoxes) to the days of Isaac Newton.

    But thanks to him, we know the answer. Look carefully at the words "at this point" in your post (as quoted here). The ball is "not moving" only for an infinitely short moment. It does not spend any length of time remaining motionless, it just slows down to zero and immediately starts accelerating in the opposite direction.

    Its speed is continuosly changing. Take any specific value of speed within a realistic range, say +1 m/s (assume the positive direction is downwards). Now ask for how long the ball will have exactly that speed. The answer will also be "for an infinitely short moment". The same goes for speed zero.

    It's like asking where a flying arrow is. Suppose for simplicity that it follows a straight line, and there is a point zero somewhere in its path. Ignore the length of the arrow - think of one point of it, say, the tip. Is there a time when the tip is at point zero?

    Yes and no.

    The tip only spends an infinitely short length of time at precisely point 0, so you could say no, it spends no time there.
    On the other hand, yes, the tip of the arrow does "show up" at point 0, and anybody doubting that is welcome to try and stand there.

    Does this help?


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  4. #3 Re: A problem with F=ma 
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    Zeno’s paradoxes have nothing to do with this problem. The problem is that Waveman28 is confusing velocity with acceleration. When the ball is stationary, it is the velocity that is 0, not its acceleration. Hence still applies to the stationary ball.
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  5. #4  
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    Yes, Waveman28 is confusing velocity with acceleration. The only thing force does is give acceleration to the body to which it is applied (neglecting any torque). In a motion like that of a projectile, one can see that the body may have acceleration even when the velocity is zero.

    Consider another situation: A ball is moving with some velocity (u) towards a batsman who hits it with his bat so that the ball returns in the opposite direction with a velocity (v). Now u and v are in opposite directions. This implies that the bat delivered some force on the ball to change the direction of its velocity. During the time in which the force is applied, the velocity decreases from u to zero and then increases from zero to v (actually it decreases considering the vectors). Thus, at some instant the velocity is zero but the force doesn't stop pushing the ball back. So there is still some acceleration to increase the ball's speed again.

    Therefore, F=ma is perfectly correct.
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  6. #5 Re: A problem with F=ma 
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    Quote Originally Posted by Leszek Luchowski
    Quote Originally Posted by Waveman28
    At this point, F=ma so the force is 9.8N. How can this be if the ball is not even moving?
    This is not a trivial question at all. This kind of questions led great minds to discuss the nature of motion, speed and time for millennia - in fact, from antiquity (see Zeno's paradoxes) to the days of Isaac Newton.

    But thanks to him, we know the answer. Look carefully at the words "at this point" in your post (as quoted here). The ball is "not moving" only for an infinitely short moment. It does not spend any length of time remaining motionless, it just slows down to zero and immediately starts accelerating in the opposite direction.

    Its speed is continuosly changing. Take any specific value of speed within a realistic range, say +1 m/s (assume the positive direction is downwards). Now ask for how long the ball will have exactly that speed. The answer will also be "for an infinitely short moment". The same goes for speed zero.

    It's like asking where a flying arrow is. Suppose for simplicity that it follows a straight line, and there is a point zero somewhere in its path. Ignore the length of the arrow - think of one point of it, say, the tip. Is there a time when the tip is at point zero?

    Yes and no.

    The tip only spends an infinitely short length of time at precisely point 0, so you could say no, it spends no time there.
    On the other hand, yes, the tip of the arrow does "show up" at point 0, and anybody doubting that is welcome to try and stand there.

    Does this help?
    That was a great explanation, thanks for helping me realise that my thinking at zero velocity also aplies to another quantity like 2m/s velocity. However, in response to others, what I am saying is that something can be accelerating whilst still stationary (for an infinitely short period of time, thanks Leszek Luchowski), and this formula only takes acceleration into account. My point then is: how can a projectile that has no velocity, hence no momentum, have the ability to exert force as described by F=ma?
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  7. #6 Re: A problem with F=ma 
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    [quote="Waveman28"]
    Quote Originally Posted by Leszek Luchowski
    My point then is: how can a projectile that has no velocity, hence no momentum, have the ability to exert force as described by F=ma?
    To answer the question as it is asked in words: an object with no velocity, even a motionless object (such as a parked car), can exert a force (pressure on the surface it is standing on).

    To answer the formula in the question: F=ma describes a relationship between three values: force, mass, and acceleration. In the case of a moving projectile, the force is the cause, the acceleration is the effect, and you seem to reason the other way round. The force is the gravitational pull the projectile is experiencing. It does not depend on speed or acceleration, it affects both, directly or indirectly.

    To answer what I think really is your problem: while flying (ignoring air drag), the projectile has a constant acceleration equal to the gravitational acceleration, i.e. 9.81 m/s2 (I assume we are on Earth on a reasonable altitude). This acceleration is constant regardless of the speed at any given moment, whether the speed is upwards, downwards or (for an infinitely short moment) zero.

    Is it getting any clearer?
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    When the ball is still rising, there is in fact no upwards force on it - all that is causing it to move upwards is its own momentum. the momentum decreases over time because there is still a downwards force on it, and as a result so does the velocity. Since the gravitational force does not depend on the upwards velocity, the velocity decreases at 9.81 m/s every second.

    So even as the velocity decreases, and passes briefly through 0, the force does not change because the acceleration is the same.

    F=ma

    a= (v-u)
    ........t

    F=m*(v-u)
    .........t

    So, to measure the acceleration, and to calculate the force, you take two readings and measure the change. Therefore, unless the acceleration is constant (in this case it is) you would take two readings for velocity and calculate the acceleration from that.

    Makes sense?

    At any one time, there is no acceleration. Acceleration can only happen between two points in time.
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    For starters F=MA is only the partial equation, the full formula eludes me at this time but I belive it involves a ^2 somewhere, as a result the common F=MA does not work in all instances, the modified formula does however (simply because even newton wasn't smart enough to predict things like lightspeed and singularity points)



    your example is flawed as well, a 1kg ball at the peak of it's upwards movement will have 0 acceleration the ball constantly has acceleration created by gravity, as it moves upwards the acceleration provided by gravity is -9.813 and as it falls the acceleration is 9.813, now evidently there has to be a point where there is no acceleration provided by gravity this is the peak of the balls upward motion

    as the ball moves upwards the -9.813 lowers the positive acceleration provided by your hand throwing the ball, at the same time gravities acceleration increases to it's norm of 9.813, so at the peak the ball has 0 acceleration meaning

    F=MA becomes F=9.81N x 0
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  10. #9  
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    Quote Originally Posted by Booms
    For starters F=MA is only the partial equation, the full formula eludes me at this time but I belive it involves a ^2 somewhere, as a result the common F=MA does not work in all instances, the modified formula does however (simply because even newton wasn't smart enough to predict things like lightspeed and singularity points)



    your example is flawed as well, a 1kg ball at the peak of it's upwards movement will have 0 acceleration the ball constantly has acceleration created by gravity, as it moves upwards the acceleration provided by gravity is -9.813 and as it falls the acceleration is 9.813, now evidently there has to be a point where there is no acceleration provided by gravity this is the peak of the balls upward motion
    Not exactly. Acceleration due to gravity is constant, and since acceleration is a vector it is always +9.813 or -9.813. It does not switch between the two.

    And, gravity does not lower the acceleration caused by the throwing of the ball - once the ball has left your hand, it is no longer being accelerated upwards, even if the accelerations are not resolved. As said above, there is no force pushing the ball up. So the only force is a downwards acceleration of 9.813 m/sē caused by gravity. Therefore, the upwards speed of the ball decreases by 9.813m/s every second. This continues below 0, so that the upward velocity is negative - i.e falling. The ball will fall faster and faster; it's speed increasing by 9.813 m/s every second.

    key points that people are missing:

    -Acceleration and force caused by gravity are both constant.

    -Acceleration is not speed. It is the rate of change of speed.


    Quote Originally Posted by Booms
    as the ball moves upwards the -9.813 lowers the positive acceleration provided by your hand throwing the ball, at the same time gravities acceleration increases to it's norm of 9.813, so at the peak the ball has 0 acceleration meaning

    F=MA becomes F=9.81N x 0
    First off, the mass is measured in kg.

    Secondly, the acceleration caused by gravity is always 9.81 m/sē.

    So the real calculation is:

    F=MA
    F=1kg x 9.81m/sē
    F=9.81N
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  11. #10  
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    Quote Originally Posted by drowsy turtle
    At any one time, there is no acceleration. Acceleration can only happen between two points in time.
    Wrong. This is like saying that "there is no speed at any one time, because speed can only happen between two points in time".
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  12. #11  
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    Quote Originally Posted by Leszek Luchowski
    Quote Originally Posted by drowsy turtle
    At any one time, there is no acceleration. Acceleration can only happen between two points in time.
    Wrong. This is like saying that "there is no speed at any one time, because speed can only happen between two points in time".
    By thw way it is calculated, acceleration requires two seperate points of time, and the length of time between them;

    a = (v-u)/t

    where a is acceleration, v is end velocity, u is start velocity and t is time.

    Alternatively, it can be calculated from the gradient of a velocity-time graph, but this would mean it is using many seperate points in time, not just one. Either way, at any given point the velocity has a defined value, so acceleration as the rate of change of velocity, does not exist.

    It's like saying if I'm travelling at 5m/s, how far do I move in 0 seconds? The answer is 0, but that doesn't mean the velocity is 0.

    So, although the acceleration still exists all the time, it can't be quantified using only one point in time.
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    This is getting quite confusing now. Would it be correct to say that at the highest point of the ball's motion when its velocity is completely zero, it cannot exert a force on anything at that exact time, contrary to F=ma?
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    Quote Originally Posted by Waveman28
    This is getting quite confusing now. Would it be correct to say that at the highest point of the ball's motion when its velocity is completely zero, it cannot exert a force on anything at that exact time, contrary to F=ma?
    For the whole of the duration, there is only one force acting on the ball - gravity.

    Forces cause acceleration.

    The ball is accelerating in a downwards direction.

    Hence the only force is gravity, which takes a constant value.
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    Quote Originally Posted by drowsy turtle
    Quote Originally Posted by Waveman28
    This is getting quite confusing now. Would it be correct to say that at the highest point of the ball's motion when its velocity is completely zero, it cannot exert a force on anything at that exact time, contrary to F=ma?
    For the whole of the duration, there is only one force acting on the ball - gravity.

    Forces cause acceleration.

    The ball is accelerating in a downwards direction.

    Hence the only force is gravity, which takes a constant value.
    Yes, but things get a bit murky if we accept Einsteins principal of equivalence and his explaination that gravity is not a force, but an acceleration. Now, an acceleration is causing an acceleration! This makes little sense.
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    Quote Originally Posted by Waveman28
    Yes, but things get a bit murky if we accept Einsteins principal of equivalence and his explaination that gravity is not a force, but an acceleration. Now, an acceleration is causing an acceleration! This makes little sense.
    Gravity is a force. It is one of the four fundermental forces of the universe along with the electromagnetic force, the weak force, and the strong nuclear force.

    Gravity causes things to accelerate it is not the object accelerating. It does this by exhibiting a force toward the center of mass.
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    Waveman28, I think you should stop thinking that the ball stops at any point. It's movement is just as continuous when it reverses direction as anywhere else during the freefall. The ball does not actually stop for any amount of time. If you plot the ball's speed over time there is no point where the curve is broken. If you take negative as velocity in the opposite direction, then the curve just goes straight through the -axis.
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  18. #17  
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    Quote Originally Posted by drowsy turtle
    Either way, at any given point the velocity has a defined value, so acceleration as the rate of change of velocity, does not exist.

    It's like saying if I'm travelling at 5m/s, how far do I move in 0 seconds? The answer is 0, but that doesn't mean the velocity is 0.

    So, although the acceleration still exists all the time, it can't be quantified using only one point in time.
    Velocity is the rate of change of position.

    Acceleration is the rate of change of velocity.

    Get some basic reading about the derivative of a function - this is what Newton used to cope with precisely the confusion you are in.

    Good luck.
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    Sorry to hijack one post... but i am just testing something - i want to see if this place has a latex installed:



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  20. #19 Re: A problem with F=ma 
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    Quote Originally Posted by Waveman28
    F=ma cannot be correct. Consider projectile motion, we throw a 1kg ball in the air and at some point it will be totally stationary in the air with no velocity. At this point, F=ma so the force is 9.8N. How can this be if the ball is not even moving? This happened the other day when I was trying to calculate the force on a moving object. What do I do in this situation? This may seem like a very elementary question but I know of no other formulas for this situation.
    Of course it must be correct. The reason why because in general (without any special relativistic intakes into this), we can still calculate to some approximation which works well under the mathematical equation itself.

    So yes, it is right.
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    Manynames, there's a stickied post on Tex in the math area.
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    Quote Originally Posted by MagiMaster
    Manynames, there's a stickied post on Tex in the math area.
    Thanks, so there is.
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  23. #22  
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    Quote Originally Posted by Leszek Luchowski
    Quote Originally Posted by drowsy turtle
    Either way, at any given point the velocity has a defined value, so acceleration as the rate of change of velocity, does not exist.

    It's like saying if I'm travelling at 5m/s, how far do I move in 0 seconds? The answer is 0, but that doesn't mean the velocity is 0.

    So, although the acceleration still exists all the time, it can't be quantified using only one point in time.
    Velocity is the rate of change of position.

    Acceleration is the rate of change of velocity.

    Get some basic reading about the derivative of a function - this is what Newton used to cope with precisely the confusion you are in.

    Good luck.
    OK, let me give you an example.

    An object is moving, at this moment in time, at 5m/s.

    What is it's acceleration?

    You can't answer, because you don't know what it's velocity was before this.

    That's what I meant.
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    Quote Originally Posted by Waveman28
    This is getting quite confusing now. Would it be correct to say that at the highest point of the ball's motion when its velocity is completely zero, it cannot exert a force on anything at that exact time, contrary to F=ma?
    If somthing is not moving than it cannot exert force (no gravity scenario), obviously. But if something is not moving it is also not crossing space, so once again, it is incable of exerting force at that infinitely small moment in time, of which is passed infintiely fast; so, in fact, the object is only capable of "force" when we bring in the notions of hypothetical time, which is genertaed by the very fact that objects move.
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    Quote Originally Posted by drowsy turtle
    OK, let me give you an example.

    An object is moving, at this moment in time, at 5m/s.

    What is it's acceleration?

    You can't answer, because you don't know what it's velocity was before this.

    That's what I meant.
    I'll turn your example around and ask how you can know that the speed "at this moment in time" is 5m/s.

    A 10-year-old could say to you:

    At this moment in time, the object is at a height of 4m.

    What is it's speed?

    You can't answer, because you don't know what it's position was before this.


    See what I mean? You have more knowledge than the 10 year old. You have already learned to think about speed at a point in time without resorting to "1 minute ago". Now try to think in the same way about acceleration at a point in time.

    Hope this helps,
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    Quote Originally Posted by drowsy turtle
    OK, let me give you an example.

    An object is moving, at this moment in time, at 5m/s.

    What is it's acceleration?

    You can't answer, because you don't know what it's velocity was before this.

    That's what I meant.
    If you gave me the velocity of 0.01 s ago, I still couldn't give you the acceleration, only a rough estimate based on some assumptions. Even if you gave me the speed at 1000 discrete points around this moment in time, I could only give you an approximation of the acceleration.
    But if you gave me the resulting force exerted on the object, and the mass of the object, I could give you the acceleration.

    Some posters need to realise that speed is relative. F=ma also works in a moving frame of reference. You could make the speed zero on any point of the trajectory of the object, and it wouldn't change it's trajectory, nor it's acceleration, nor the forces working on it (as long as the frame of reference isn't accelerating and you assume Newtonian mechanics to be valid).
    Quote Originally Posted by Booms
    ... the acceleration provided by gravity is -9.813...
    I just had to comment on this: g varies roughly between and , depending on your location on earth (centripetal force, rock formations, height above sea level...). Using a more accurate number is pointless, unless you made local measurements for an extremely sensitive application.
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    Quote Originally Posted by Leszek Luchowski
    Quote Originally Posted by drowsy turtle
    OK, let me give you an example.

    An object is moving, at this moment in time, at 5m/s.

    What is it's acceleration?

    You can't answer, because you don't know what it's velocity was before this.

    That's what I meant.
    I'll turn your example around and ask how you can know that the speed "at this moment in time" is 5m/s.

    A 10-year-old could say to you:

    At this moment in time, the object is at a height of 4m.

    What is it's speed?

    You can't answer, because you don't know what it's position was before this.


    See what I mean? You have more knowledge than the 10 year old. You have already learned to think about speed at a point in time without resorting to "1 minute ago". Now try to think in the same way about acceleration at a point in time.

    Hope this helps,
    L.
    I didn't say it doesn't exist (or didn't mean to).

    I'm saying it can't be measured without two points. Someone disagreed with me misunderstanding what I said, and that's how this started.
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    Quote Originally Posted by drowsy turtle
    I didn't say it doesn't exist (or didn't mean to).

    I'm saying it can't be measured without two points. Someone disagreed with me misunderstanding what I said, and that's how this started.
    And you are wrong. It is perfectly possible to measure the acceleration in a single point. This is exactly what an accelerometer does.
    Measuring the velocity is often more difficult.
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  29. #28  
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    Quote Originally Posted by Bender
    Quote Originally Posted by drowsy turtle
    I didn't say it doesn't exist (or didn't mean to).

    I'm saying it can't be measured without two points. Someone disagreed with me misunderstanding what I said, and that's how this started.
    And you are wrong. It is perfectly possible to measure the acceleration in a single point. This is exactly what an accelerometer does.
    Use an accelerometer and you will notice a lag between a change in acceleration and a change in reading.

    Acceleration is the rate of change of velocity, so it requires a period of time in which the rate is measured. Read my previous example and tell me the acceleration.

    Quote Originally Posted by Bender
    Measuring the velocity is often more difficult.
    Acceleration is derived from velocity, so no, not really.
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    Quote Originally Posted by drowsy turtle
    Quote Originally Posted by Bender
    Quote Originally Posted by drowsy turtle
    I didn't say it doesn't exist (or didn't mean to).

    I'm saying it can't be measured without two points. Someone disagreed with me misunderstanding what I said, and that's how this started.
    And you are wrong. It is perfectly possible to measure the acceleration in a single point. This is exactly what an accelerometer does.
    Use an accelerometer and you will notice a lag between a change in acceleration and a change in reading.

    Acceleration is the rate of change of velocity, so it requires a period of time in which the rate is measured. Read my previous example and tell me the acceleration.

    Quote Originally Posted by Bender
    Measuring the velocity is often more difficult.
    Acceleration is derived from velocity, so no, not really.
    I guess you don't know how most accelerometers work. They are usually force sensors with a small inertial mass attached, using F/m=a to get to the acceleration.
    Getting the acceleration from velocity is an unstable process as it requires the derivation of often noisy data, blowing up any error. Most industrially used "velocity" sensors are actually position sensors or counters.
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  31. #30  
    Reptile Dysfunction drowsy turtle's Avatar
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    Quote Originally Posted by Bender
    Quote Originally Posted by drowsy turtle
    Quote Originally Posted by Bender
    Quote Originally Posted by drowsy turtle
    I didn't say it doesn't exist (or didn't mean to).

    I'm saying it can't be measured without two points. Someone disagreed with me misunderstanding what I said, and that's how this started.
    And you are wrong. It is perfectly possible to measure the acceleration in a single point. This is exactly what an accelerometer does.
    Use an accelerometer and you will notice a lag between a change in acceleration and a change in reading.

    Acceleration is the rate of change of velocity, so it requires a period of time in which the rate is measured. Read my previous example and tell me the acceleration.

    Quote Originally Posted by Bender
    Measuring the velocity is often more difficult.
    Acceleration is derived from velocity, so no, not really.
    I guess you don't know how most accelerometers work. They are usually force sensors with a small inertial mass attached, using F/m=a to get to the acceleration.
    Getting the acceleration from velocity is an unstable process as it requires the derivation of often noisy data, blowing up any error. Most industrially used "velocity" sensors are actually position sensors or counters.
    Ah, good point. I withdraw my claims.
    "The major difference between a thing that might go wrong and a thing that cannot possibly go wrong is that when a thing that cannot possibly go wrong goes wrong it usually turns out to be impossible to get at or repair." ~ Douglas Adams
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  32. #31  
    Forum Ph.D. Leszek Luchowski's Avatar
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    Quote Originally Posted by drowsy turtle
    Use an accelerometer and you will notice a lag between a change in acceleration and a change in reading.
    Show me any measuring instrument that does not have some kind of delay between the quantity it measures and its output.

    Quote Originally Posted by drowsy turtle
    Acceleration is the rate of change of velocity, so it requires a period of time in which the rate is measured.
    Why don't you say the same about velocity, which is the rate of change of position?
    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
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  33. #32  
    Reptile Dysfunction drowsy turtle's Avatar
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    Quote Originally Posted by drowsy turtle
    I withdraw my claims.
    "The major difference between a thing that might go wrong and a thing that cannot possibly go wrong is that when a thing that cannot possibly go wrong goes wrong it usually turns out to be impossible to get at or repair." ~ Douglas Adams
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