Deriving the rate time equation for a Jerk (J)

[~christina~]

Homework Statement

Automotive engineers refer to the time rate change of acceleration as the Jerk.
Assuem an object moves in 1 dimension such that it's Jerk is constant.

a) determine expressions for it's
1. acceleration ax(t)
2. velocity vx(t)
3. position x(t)

given initial acceleration, velocity, and positio are axi, vxi, xi respectively

b) show that ax^2= axi^2 + 2J(vx-vxi)

Homework Equations

not sure because time rate of acceleration = Jerk...but what is the time unit..is it just sec?

I know that the time rate change for velocity is acceleration though..

a= v/t

Not sure about the position

The Attempt at a Solution

including the equation attempt above..the whole problem is to determine the equation..I'm just not sure how to put it together

I know that the problem is under deriving equations from calculus though just I know that if you get the derivative of the position then you get velocity eqzn and if you derive that you get the acceleration.
And if you get the integral of the acceleration then you get the velocity equation and if you get the integral of that you get the position function once again..

other than that I just don't know how to relate equation deriving to a "Jerk"

I have problems with deriving eqzns..


for b.) I'd don't know what they want me to do with it...exactly...

Thanks

 

[cepheid]

First, of course the standard units of time are seconds, no matter which unit system you are using.

Saying that the jerk is the rate of change of acceleration with time is the same as saying that the jerk is the derivative of the acceleration with respect to time. In other words:

velocity is the first derivative of position
acceleration is the second derivative of position
jerk is the third derivative of position.

Based on this, I think you can see that:

J(t) = \textrm{const.} = J_x

a_x(t) = \int J_x \, dt
v_x(t) = \int a_x(t) \, dt
x(t) = \int v_x(t) \, dt

Now it's just a matter of computation.

 

[~christina~]

check

But what do I compute?? and how do I prove the eqzn

b) show that ax^2= axi^2 + 2J(vx-vxi)

how do I show that??

When You say compute don't you have to have a original equation to compute from?

Is it...since Jx= constant

ax(t)=\int Jx dt = Jt

then vx(t)=\int ax(t)dt= \int Jtdt = J/2*t^2

then x(t)= \int vx(t)dt= \int J/2* t^2= J/6*t^3


Is this fine?

 

[learningphysics]

But what do I compute?? and how do I prove the eqzn

b) show that ax^2= axi^2 + 2J(vx-vxi)

how do I show that??

When You say compute don't you have to have a original equation to compute from?

Is it...since Jx= constant

ax(t)=\int Jx dt = Jt

then vx(t)=\int ax(t)dt= \int Jtdt = J/2*t^2

then x(t)= \int vx(t)dt= \int J/2* t^2= J/6*t^3


Is this fine?

add constants when you take the integrals.

 

[~christina~]

Okay Thank You. But what about the part B?

b) show that ax^2= axi^2 + 2J(vx-vxi)

how do I show that??

 

[learningphysics]

Okay Thank You. But what about the part B?

b) show that ax^2= axi^2 + 2J(vx-vxi)

how do I show that??

When you add the constants, this will make sense.

eg:

ax = Jt + axi

vx = (J/2)t^2 + axi*t + vxi etc...

 

[~christina~]

hm I thought you meant the constant c at the end of the integral...ex usually X+C or etc.

So do you mean that I add...it like this...

ax(t)=\int Jxdt = Jt + axi

then vx(t)=\int ax(t)dt=\int Jt + axi= J/2*t^2 + axi + vxi

then x(t)=\int vx(t)dt=\int J/2*t^2 + axi +vxi= J/6*t^3 +axi + vxi + xi

b) show that ax^2= axi^2 + 2J(vx-vxi)

now..what do I add?

You said that it was:
When you add the constants, this will make sense.

eg:

ax = Jt + axi

vx = (J/2)t^2 + axi*t + vxi etc...


Am I going in the right direction with this??
b/c would I use the end equation for x or add them all together?

 

[learningphysics]

hm I thought you meant the constant c at the end of the integral...ex usually X+C or etc.

So do you mean that I add...it like this...

ax(t)=\int Jxdt = Jt + axi

then vx(t)=\int ax(t)dt=\int Jt + axi= J/2*t^2 + axi + vxi

that's not right. it should be J/2*t^2 + axi*t + vxi

 

[~christina~]

see below

 

[~christina~]

Oh..I forgot that..well is right now?

x(t)=\int vx(t)dt= J/6*t^3+ axi/2*t^2+ vxi*t + xi

but how do I get what they want me to get?? since they have ax and axi and etc in the equation?

 

[learningphysics]

Oh..I forgot that..well is right now?

x(t)=\int vx(t)dt= J/6t^3+ axi/2*t^2+ vxi*t + xi

Yeah, that looks right.

you should be able to do part b) using your vx and ax formulas.

 

[learningphysics]

Oh..I forgot that..well is right now?

x(t)=\int vx(t)dt= J/6*t^3+ axi/2*t^2+ vxi*t + xi

but how do I get what they want me to get?? since they have ax and axi and etc in the equation?

work out ax^2 using your formula for ax.

 

[~christina~]

I'm stumped...I really don't know where I get ax^2 from that equation...
do I use originally what I got for the indef interals of each?
thus

ax= Jt +axi
vx= J/2*t^2 + axi*t + vxi
x= J/6*t^3+ axi/2*t^2+ vxi*t + xi

I'm not sure do I add them together...all of the equations??

 

[learningphysics]

I'm stumped...I really don't know where I get ax^2 from that equation...
do I use originally what I got for the indef interals of each?
thus

ax= Jt +axi
vx= J/2*t^2 + axi*t + vxi
x= J/6*t^3+ axi/2*t^2+ vxi*t + xi

I'm not sure do I add them together...all of the equations??

ax = Jt + axi

ax^2 = J^2t^2 + 2Jt(axi) +axi^2 = 2J(J/2*t^2 + axi*t) + axi^2

Do you see how you can get vx-vxi in the above?

 

[~christina~]

Yes since you can plug in

vx= J/2*t^2 + axi*t + vxi
vx-vxi= J/2*t^2 + axi*t so you substitute that into

ax^2 = J^2t^2 + 2Jt(axi) +axi^2 = 2J(J/2*t^2 + axi*t) + axi^2

and get...

ax^2 = J^2t^2 + 2Jt(axi) +axi^2 = 2J(vx-vxi) + axi^2


I get that now...I got confused from your previous post that said I had to add constants..so it was simply square the original ax...

 

[learningphysics]

Yes since you can plug in

vx= J/2*t^2 + axi*t + vxi
vx-vxi= J/2*t^2 + axi*t so you substitute that into

ax^2 = J^2t^2 + 2Jt(axi) +axi^2 = 2J(J/2*t^2 + axi*t) + axi^2

and get...

ax^2 = J^2t^2 + 2Jt(axi) +axi^2 = 2J(vx-vxi) + axi^2


I get that now...I got confused from your previous post that said I had to add constants..so it was simply square the original ax...

Yeah, sorry... I just meant to include constants (axi, vxi, xi).

Anyway, that looks right. good job.

 

[~christina~]

yay!

thanks for all your help