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Missing Velocity Paradox

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Guns N. Roses

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Missing Velocity Paradox
« on: May 01, 2007, 07:23:15 PM »
Ok its not actually a paradox but here goes,

Lets say we have a distance function s(t)=-|t+3|

The derivative at 3 is undefined at 3 because it is an absolute value

The interval is from 0 to 6. By the mean value theorem some point c has to have a derivative of 0.

Therefore the velocity at 3 is 0, which seems logical, but that means the derivative at 3 is 0 which it isnt!

What do you think?
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sokarul

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Re: Missing Velocity Paradox
« Reply #1 on: May 01, 2007, 08:19:41 PM »
umm the derivative is the change in speed.  So the derivative will not be zero since the equation is linear.    The change in speed is always negative 1.  Nothing abnormal. 
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Bushido

Re: Missing Velocity Paradox
« Reply #2 on: May 02, 2007, 04:19:06 AM »
Quote from: Guns N. Roses on May 01, 2007, 07:23:15 PM
Ok its not actually a paradox but here goes,

Lets say we have a distance function s(t)=-|t+3|

The derivative at 3 is undefined at 3 because it is an absolute value

The interval is from 0 to 6. By the mean value theorem some point c has to have a derivative of 0.

Therefore the velocity at 3 is 0, which seems logical, but that means the derivative at 3 is 0 which it isnt!

What do you think?

First of all, I think that it should be s(t) = |-t+3|. Second, and most importantly, the mean value theorem has the following assumptions:
1) s(t) has to be continuous on the segment [a,b];
2) s(t) has to have a derivative on the interval (a,b).
These conditions are necessary. Since 2) is not fullfiled for t = 3 which is in (0, 6), the theorem is not applicable.
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #3 on: May 02, 2007, 06:26:49 AM »
sorry, i mean s(t)=-|x-3|+3
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #4 on: May 02, 2007, 06:27:41 AM »
Then what IS the velocity at 3?
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Durdan

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Re: Missing Velocity Paradox
« Reply #5 on: May 02, 2007, 07:57:10 AM »
veloceration
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sokarul

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Re: Missing Velocity Paradox
« Reply #6 on: May 02, 2007, 09:33:05 AM »
Quote from: Guns N. Roses on May 02, 2007, 06:26:49 AM
sorry, i mean s(t)=-|x-3|+3
The derivative is still negative one. 
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Quote from: Shifter on July 05, 2020, 09:53:30 AM
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #7 on: May 02, 2007, 01:45:56 PM »
How can the derivative always be negative one? From 0 to 3 it is one, from 3 to 6 it is negative one, but what is it AT 3? S(t) is not just a strait line
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sokarul

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Re: Missing Velocity Paradox
« Reply #8 on: May 02, 2007, 01:56:12 PM »
O right.  Its 1 as  x aproches negative 3 and then negative 1 on.  There is nothing strange though.
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Quote from: Shifter on July 05, 2020, 09:53:30 AM
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #9 on: May 02, 2007, 02:05:39 PM »
Exactly, but I need to know what the velocity is AT 3.
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sokarul

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Re: Missing Velocity Paradox
« Reply #10 on: May 02, 2007, 03:06:57 PM »
my calculator says 3 with a derivative of zero. 
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Quote from: Shifter on July 05, 2020, 09:53:30 AM
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #11 on: May 02, 2007, 05:11:02 PM »
Thats because it is a calculator.

http://members.aol.com/organichem/my_pages/absvalder.html

lim(h->0) of (|x+h|-|x|)/h  is the function to find the derivative at x

lim(h->0) of (|0+h|-|0|)/h

     =|h|/h

Split it into two limits

lim(h->0+) of h/h=1

lim(h->0-) of -h/h=-1

There fore the limit doesn't exist at the point of an absolute value.
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sokarul

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Re: Missing Velocity Paradox
« Reply #12 on: May 02, 2007, 05:41:40 PM »
I don't think you are disproving calculus.  My calculator gets all the other values right, why wouln't it get x=-3 right?
« Last Edit: May 02, 2007, 05:44:04 PM by sokarul »
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Quote from: Shifter on July 05, 2020, 09:53:30 AM
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #13 on: May 02, 2007, 06:11:42 PM »
Im NOT disproving calculus, Im wondering what the  value would be at (positive, not negative) 3. A caclulator doesn't get everything right when it comes to this stuff. It just picks 0 for that spot.
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qwe

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Re: Missing Velocity Paradox
« Reply #14 on: May 02, 2007, 10:26:41 PM »
Quote
The interval is from 0 to 6. By the mean value theorem some point c has to have a derivative of 0.
the mean value theorem only applies if it is continuous on the interval

since s' is undefined at 3, it's not continuous
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Bushido

Re: Missing Velocity Paradox
« Reply #15 on: May 03, 2007, 02:49:34 AM »
Quote from: Guns N. Roses on May 02, 2007, 06:26:49 AM
sorry, i mean s(t)=-|x-3|+3

The expression under the absoulte value is non - negative for:

t - 3 ≥ 0 <=> t ≥ 3

Thus, we can write the given function for two disjunctive intervals:

s(t) = -(t - 3) + 3 = -t + 6,   t ≥ 3

and

s(t) = -[-(t-3)] + 3 = t,        t ≤ 3

The derivative of this function is:

s'(t) = 1,   t ≤ 3
s'(t) = -1,  t ≥ 3

Te left hand and the right hand derivative at t = 3 are different, so, the derivative at t = 3 does not exist. Thus, the conditions from the mean value theorem are not fullfilled (see my previous post). End of discussion!
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Guns N. Roses

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Re: Missing Velocity Paradox
« Reply #16 on: May 03, 2007, 05:58:51 AM »
Yes I understand what you said about the mean value theorem, but thinking in a physical sense it doesn't make any. What I'm wondering is if (remember this is a distance function) you can have no velocity at a point. Or if you can skip from -1 to 1 without going through anything between.
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Bushido

Re: Missing Velocity Paradox
« Reply #17 on: May 03, 2007, 07:41:55 AM »
The problem is that you gave an example of a step function for the velocity. This does not mean that one exists in reality. The acceleration of the particle from the following example can be expressed by Dirac's delta-function:

a(t) = -2*δ(t - 3)

Now, these kinds of accelerations don't exist in reality, but are good approximation for impulse forces, like in the case of collisions.
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Spherical Earth Society Leader

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Re: Missing Velocity Paradox
« Reply #18 on: June 11, 2007, 05:31:43 PM »
The equation should have been:

a{6+2x[b-3|3+c|656735623453457x(q+304|36|4)+35xc]+98swf+9}fv~6.456

whereas:
a= +345 or -785
x= +632, +35562, or -12
b= +356734563457 or -1
c= 0, 1, 2, 3, or -3124562347245624574568
q= any nonrelative integers of which the number is not zero
s= any relative integers of which the number is zero or into infinity above and infinity below
w= -34564
f= +34623, or -735623463568723456346236523452734562347.345634111111111111...
v= 0 and to above infinity

and whereas:
~ is the differential position of the axis of the odontoid of the parabolic equation previously stated.
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BOGWarrior89

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Re: Missing Velocity Paradox
« Reply #19 on: June 11, 2007, 06:47:41 PM »
Quote from: Guns N. Roses on May 03, 2007, 05:58:51 AM
Yes I understand what you said about the mean value theorem, but thinking in a physical sense it doesn't make any. What I'm wondering is if (remember this is a distance function) you can have no velocity at a point.

You've never been stopped?

Quote from: Guns N. Roses on May 03, 2007, 05:58:51 AM
Or if you can skip from -1 to 1 without going through anything between.

You do go through things between.
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