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Jerk - Derivative of Acceleration


Date: 03/16/2001 at 15:24:25
From: Liz
Subject: Mathematical term that starts with the letter J

I just need a mathematical term that begins with the letter J, and 
its definition and a high school level explanation of the term. 
Possibly some examples and pictures to go along with the definition.

Thank you.


Date: 03/16/2001 at 16:16:58
From: Doctor Ian
Subject: Re: Mathematical term that starts with the letter J

Hi Liz,

A mathematical term beginning with 'j' that can be explained at 
high-school level is 'jerk'. 

Let's say I toss a ball into the air and let it bounce on the floor.  
I could plot the height ('position') above the floor as a function of 
time:

   |
   |        p
   |    p       p
   |  p           p
   |
   | p             p
   | 
   |
   |p               p
   |                   p
   |
   |                 
   +-----------------p----


(I apologize for the clunky nature of the graphics.  There's only so 
much you can do with a keyboard and a fixed-width font.)   

The derivative of this function, called the 'velocity', tells me how 
the position is changing at any given time. When the ball is going up, 
the velocity is positive. When the ball is going down, the velocity is
negative. For a brief period when the ball is at the top of its arc, 
the velocity is zero:


   |
   |        p
   |    p       p
   |  p           p
   |
   | p             p
   | 
   |v                  v
   |p               p
   |                   p
   |    v
   |                   
   +--------v--------p----
              
                v
                
                    v
              

The derivative of the velocity, called the acceleration, tells me how
the velocity is changing. In this case, the change in velocity is 
constant per unit time, until the ball hits the floor, at which time 
it becomes very positive:


   |
   |        p
   |    p       p
   |  p           p
   |                   a
   | p             p
   |                    
   |v                  v
   |p               p
   |                   p
   |    v
   |                   
   +--------v--------p----
              
                v
    aaaaaaaaaaaaaaaaaa            
                    v

So what does the derivative of the acceleration, called the 'jerk', 
tell you?  It tells you how the acceleration is changing. In this 
case, there is no change in acceleration until the ball hits the 
floor:
 
   |
   |        p
   |    p       p
   |  p           p
   |                   a
   | p             p
   |                    
   |v                  v
   |p               p
   |                   p
   |    v            j
   |                   
   +jjjjjjjjvjjjjjjjjpjjjj
              
                v
    aaaaaaaaaaaaaaaaaa            
                    v

Basically, if you plot jerk as a function of time, the intervals where 
it is non-zero tell you when new forces come into play. 

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus
High School Physics/Chemistry

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