Mechanics (Components of Acceleration)

[roam]

Homework Statement

The following is a solved problem:

http://img600.imageshack.us/img600/6503/questiono.jpg

I can't understand how they worked out these two components.

Homework Equations

\vec{a}(t) = \frac{dv}{dt}\vec{T} + v \frac{d \vec{T}}{dt}

Where

\vec{T}(t) = \frac{d \vec{r}(t)/dt}{||d \vec{r}(t)/dt||} = \frac{\vec{v}(t)}{v(t)}

The Attempt at a Solution

In the equation I posted above, I believe the first term is the tangential component of acceleration and the second one is the normal component. I want first to find the tangential component:

\vec{v}(t) = d(t^2 \vec{i} + 3t+2 \vec{j})/dt = 2t+3

|| \vec{v}(t) || = \sqrt{4t^2+9}

So substituting in

\frac{dv}{dt}\vec{T} = \frac{d (\vec{r}(t)/dt)}{dt} . \frac{d \vec{r}(t)}{dt}.\frac{1}{||\vec{r}(t)||}

2.(2t+3).\frac{1}{\sqrt{4t^2+9}}

= \frac{4t+6}{\sqrt{4t^2+9}}

But this is wrong. The correct answer must be

\frac{4t}{\sqrt{4t^2+9}}

What did I do wrong? How can I get to the correct answer? Any guidance is greatly appreciated.

 

[Quinzio]

Homework Statement

The following is a solved problem:

http://img600.imageshack.us/img600/6503/questiono.jpg

I can't understand how they worked out these two components.

Homework Equations

\vec{a}(t) = \frac{dv}{dt}\vec{T} + v \frac{d \vec{T}}{dt}

Where

\vec{T}(t) = \frac{d \vec{r}(t)/dt}{||d \vec{r}(t)/dt||} = \frac{\vec{v}(t)}{v(t)}

The Attempt at a Solution

In the equation I posted above, I believe the first term is the tangential component of acceleration and the second one is the normal component. I want first to find the tangential component:

\vec{v}(t) = d(t^2 \vec{i} + 3t+2 \vec{j})/dt = 2t+3

|| \vec{v}(t) || = \sqrt{4t^2+9}

So substituting in

\frac{dv}{dt}\vec{T} = \frac{d (\vec{r}(t)/dt)}{dt} . \frac{d \vec{r}(t)}{dt}.\frac{1}{||\vec{r}(t)||}

2.(2t+3).\frac{1}{\sqrt{4t^2+9}}

= \frac{4t+6}{\sqrt{4t^2+9}}

But this is wrong. The correct answer must be

\frac{4t}{\sqrt{4t^2+9}}

What did I do wrong? How can I get to the correct answer? Any guidance is greatly appreciated.

You have speed: \vec v= \frac{dr}{dt}=(2t,3)
and acceleration: \vec a= \frac{dv}{dt}=(2,0)

then you've got to find the projection of \vec a onto \vec v:
\vec a_t = \frac{(\vec a \cdot \vec v)}{||\vec v||^2}\vec v= \frac{1}{4t^2+9}(8t^2,12t)
The normal component of acceleration is then:
\vec a_n= \vec a - \vec a_t

The given answers seem to be wrong.