What Are the Kinematics of a Particle Moving Along a Curve?

[roadrunner]

Homework Statement

The position of a particle at time t is given by
r(t) =<t,t^2,(2/3)t^3>

(a) Find the velocity v, speed |v| and acceleration a of the particle, and the
curvature k of its path, as a function of t.


(b) Find the arc length function s(t), and the arc length of the curve as t increases
from t = 0 to t = 3.

(c) At the point <1,1,2/3>, find the unit tangent, principal normal and binormal
vectors T, N and B, and find the curvature k and radius

Homework Equations

*NOTE* | | arround somethign mean the magnitude of the vector not absolute value
IE |2i,3j,4k| = sqrt(4+9+16)

T(t)= r'(t)
       |r'(t)|

k=|r'(t)Xr"(t)|
       |r'(t)|^3

N(t)=T'(t)
      |T'(t)|

T(t)=r'(t)
     |r'(t)|

B(t)=T(t)XN(t)

r'(t)=v
r"(t)=v'(t)=a

The Attempt at a Solution

a) r(t) =<t,t^2,(2/3)t^3 (given)
v= r'(t)=1 i, 2t j, 2t^2 k
a= r"(t)=v'(t)=0 i, 2 j, 4t k

|v|=|r'(t)|=sqrt(1+4t^2+4t^4)=(2t^2+1)=|V|

k=|r'(t)xr"(t)|
     |r'(t)|^3

r'(t) X r"(t)= | i    j    k  |
             |1  2t   2t^2  |  =[B](4t^2)i, -4t j, 2 k[/B]
             |0   2   4t    | 

|r'(t) X r"(t)|= sqrt(16t^4+16t^2+4)
=2sqrt(4t^4+4t^2+1)
=2sqrt(2t^2+1)^2 
= [B]2(2t^2+1)[/B]

k= 2(2t^2+1)
   (2t^2+1)^3

[B]= 2
(2t^2+1)^2[/B]

did i do this right?

s(t)=(integral form a to b of) |r'(t)|
s(t)=(int of)2t^2+1

integral form 0 to 3 is (2/3)t^3+t = (2/3)3^3 +3 = 18+3=21?


c)At the point <1,1,2/3>, find the unit tangent, principal normal and binormal
vectors T, N and B, and find the curvature k and radius

my attempt...

T(t)=r'(t)             r'(t)=1 i, 2t j 2t^2 k
     |r'(t)|             |r'(t)|=2t^2+1

T(t)=(1/(2t^2+1)) i + ((2t)/(2t^2+1)) j + ((t^2)/(2t^2+1)) k

the point 1,1,2/3 means t=1
[B]T(1)=1/3 i + 2/3 j + 1/3 k[/B]

CORRECT??!



IM STUCK i can't find T'(t)

but i found |T'(t)| with

|T'(t)|=|r'(t) X r"(t)|      =       2(2t^2+1)          =   2
             |r'(t)|^2              (2t^2+1)            2t^2+1

also i found k with the solution from part a


k= 2
(2t^2+1)^2

k=2/9?

and once i get that T'(t) i can easy use B(t)=T(t) X N(t) to finish the problem

so please point out any errors and help me figure out how to get T'(t) :D

 

[NonAbelian]

Your v x a is incorrect

 

[roadrunner]

look better now?

 

[roadrunner]

ok i fixed it, does everythign else look k correct? (b also)? 'cuse I am running onto problems on C

 

[roadrunner]

dont mean to be pushy but i need to sleep soon, can any1 shed some light please?

 

[roadrunner]

bed in 20 minutes...any1?

 

[roadrunner]

bump?