Calculating Normal Vectors for Particle Motion

[BennyT]

Homework Statement

[/B]
This problem is from Jon Rogawski's Calculus-Early Transcendentals
At a certain moment, a moving particle has velocity v={2,2,-1} and a={0,4,3}. Find T, N and the decomposition of a into tangential and normal components.

Homework Equations

ANYTHING IN [ ] REPRESENTS A LENGTH OF A QUANTITY and . stands for dot product
T[/B](t)=v(t)/[v(t)]
N(t)=T '(t)/[T '(t)]
an=√([a]^2-at^2)
at=a.T
an=a.N=√([a]^2-at^2)

The Attempt at a Solution

So I'm trying to find an, the normal component of acceleration, but by using the two definitions and equations found in the book I am coming to different values. So I first calculated T={2/3,2/3,-1/3}, and then N={0,4/5,3/5}. Then I calculated at=5/3 and then I calculated an by the first equation (an=√([a]^2-at^2)) to equal an=(10/3)(√2), but using the second equation I get an=5. What am I doing wrong? And also, do T and N always have to be orthogonal? This isn't a graded assignment or anything, but it's causing me frustration. Thanks for your time.

 

[HallsofIvy]

You appear to be assuming that the acceleration vector is itself normal to the tangent vector. That is not necessarily true because you are not told that the acceleration is constant.

 

[BennyT]

Ok, so is this reply in response to whether T and N are always orthogonal? Also, for a problem such as the one above, what is the best way to calculate N? Thank you for your response.

 

[SammyS]

Ok, so is this reply in response to whether T and N are always orthogonal? Also, for a problem such as the one above, what is the best way to calculate N? Thank you for your response.

I'm sure it wasn't about whether they are orthogonal. They definitely are.

Vector T is in the direction of motion, you did that correctly.

Vector N is perpendicular to vector T, and is in the plane of the particle's motion. Basically it's in direction of the component of acceleration vector, a, that's perpendicular to vector T .