Solving for tangential and normal components of acceleration

[Franzia12]

Hi all, I'm new and hope this is the correct format for things!

I was given a position vector r(t) = ti + 2e^tj+e^2tk and was asked to find tangential and normal components of acceleration. I've ended up with 2e^t(1+2e^2t)/sqrt(1+e^2t) as my tangential component but am unsure how simplified that is or if it is even correct. Then I proceeded to compute the normal and ended up with 2sqrt[(e^4t-e^2t+e^t)/(1+e^2t)].

All of the e functions have been messing with me and I could use some help please!

 

[pka]

Given \(\bf{r(t)}\) the tangential component of acceleration \(\bf{a_{T}=\dfrac{r'(t)\cdot r''(t)}{\|r'(t)\|}}\)
and normal components of acceleration \(\bf{a_{N}=\dfrac{\|r'(t)\times r''(t)\|}{\|r'(t)\|}}\)

In your case \(\bf{r(t) = t~i + 2e^t~j+e^{2t}~k}\) , then \(\bf{r'(t)=i+2e^t~j+2e^{2t}~k}\) and \(\bf{r''(t)=2e^t~j+4e^{2t}~k}\)
Can you finish?

 

[Franzia12]

Given \(\bf{r(t)}\) the tangential component of acceleration \(\bf{a_{T}=\dfrac{r'(t)\cdot r''(t)}{\|r'(t)\|}}\)
and normal components of acceleration \(\bf{a_{N}=\dfrac{\|r'(t)\times r''(t)\|}{\|r'(t)\|}}\)

In your case \(\bf{r(t) = t~i + 2e^t~j+e^{2t}~k}\) , then \(\bf{r'(t)=i+2e^t~j+2e^{2t}~k}\) and \(\bf{r''(t)=2e^t~j+4e^{2t}~k}\)
Can you finish?

Sorry, I probably should have elaborated. I did find my r' and r'' functions and even went on to calculate my tangential and normal components. After attempting it again my answer is different from before and now I have (4e^2t+8e^4t)/(sqrt(8e^4t+4e^2t+1). Can that be simplified more? Is that even correct?

 

[Deleted member 4993]

Sorry, I probably should have elaborated. I did find my r' and r'' functions and even went on to calculate my tangential and normal components. After attempting it again my answer is different from before and now I have (4e^2t+8e^4t)/(sqrt(8e^4t+4e^2t+1). Can that be simplified more? Is that even correct?

What is it that you have - normal component or tangential component or total acceleration? Remember that acceleration is a vector; thus it has magnitude and direction.