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hey everyone,
The title of this thread explains what I'm trying to do. I want to find an equation that can represent a jerk. A jerk is a change in acceleration with respect to time. What I really want though is an equation for a jerk that doesn't have a time variable in it. I think I know the answer but I want people to back up my answer so here we go.
We know that this is true
j=j
A jerk is equal to a jerk... this really doesn't tell us anything about a jerk. So we now integrate by dt and we get
\int jdt= jt+C_1
The units of this equation are equal to an acceleration so this equation must be this
jt+a_i=a_f
Of course, we can keep integrating till we get down to distances so i'll show those now without doing the work.
\frac{jt^2}{2}+a_it+v_i=v_f
\frac{jt^3}{6}+\frac{a_it^2}{2}+v_it=d
Ok so now I have three equations that I could at least try solving for a jerk but there is a problem. All these equations involve time. The next logical thing that I can do is to substitute "t" into an equation, but first I need an equation to solve for time. I'll use the first equation.
jt+a_i=a_f
jt=a_f-a_i
t=\frac{a_f-a_i}{j}
Now I'll substitute that into the second equation.
\frac{jt^2}{2}+a_it+v_i=v_f
\frac{j(a_f-a_i)^2}{2j^2}+\frac{a_i(a_f-a_i)}{j}+v_i=v_f
then I simplify and try to get j on one side
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{a_fa_i-a_i^2}{j}+v_i=v_f
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{a_fa_i-a_i^2}{j}=v_f-v_i
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{2a_fa_i-2a_i^2}{2j}=v_f-v_i
\frac{a_f^2-2a_fa_i+a_i^2+2a_fa_i-2a_i^2}{2j}=v_f-v_i
\frac{a_f^2-a_i^2}{2j}=v_f-v_i
\frac{a_f^2-a_i^2}{v_f-v_i}=2j
\frac{a_f^2-a_i^2}{2(v_f-v_i)}=j
So the final equation I get is this
j=\frac{a_f^2-a_i^2}{2(v_f-v_i)}
but is this right? Let's check the units. On top we have meters^2 per second^4 and on the bottom we have meters per second
\frac {m^2}{s^4}* \frac{s}{m}= \frac {m}{s^3}
Well at least the units are right but is this equation right?
Thanks for helping me out!
The title of this thread explains what I'm trying to do. I want to find an equation that can represent a jerk. A jerk is a change in acceleration with respect to time. What I really want though is an equation for a jerk that doesn't have a time variable in it. I think I know the answer but I want people to back up my answer so here we go.
We know that this is true
j=j
A jerk is equal to a jerk... this really doesn't tell us anything about a jerk. So we now integrate by dt and we get
\int jdt= jt+C_1
The units of this equation are equal to an acceleration so this equation must be this
jt+a_i=a_f
Of course, we can keep integrating till we get down to distances so i'll show those now without doing the work.
\frac{jt^2}{2}+a_it+v_i=v_f
\frac{jt^3}{6}+\frac{a_it^2}{2}+v_it=d
Ok so now I have three equations that I could at least try solving for a jerk but there is a problem. All these equations involve time. The next logical thing that I can do is to substitute "t" into an equation, but first I need an equation to solve for time. I'll use the first equation.
jt+a_i=a_f
jt=a_f-a_i
t=\frac{a_f-a_i}{j}
Now I'll substitute that into the second equation.
\frac{jt^2}{2}+a_it+v_i=v_f
\frac{j(a_f-a_i)^2}{2j^2}+\frac{a_i(a_f-a_i)}{j}+v_i=v_f
then I simplify and try to get j on one side
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{a_fa_i-a_i^2}{j}+v_i=v_f
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{a_fa_i-a_i^2}{j}=v_f-v_i
\frac{a_f^2-2a_fa_i+a_i^2}{2j}+\frac{2a_fa_i-2a_i^2}{2j}=v_f-v_i
\frac{a_f^2-2a_fa_i+a_i^2+2a_fa_i-2a_i^2}{2j}=v_f-v_i
\frac{a_f^2-a_i^2}{2j}=v_f-v_i
\frac{a_f^2-a_i^2}{v_f-v_i}=2j
\frac{a_f^2-a_i^2}{2(v_f-v_i)}=j
So the final equation I get is this
j=\frac{a_f^2-a_i^2}{2(v_f-v_i)}
but is this right? Let's check the units. On top we have meters^2 per second^4 and on the bottom we have meters per second
\frac {m^2}{s^4}* \frac{s}{m}= \frac {m}{s^3}
Well at least the units are right but is this equation right?
Thanks for helping me out!
