Angle b/w Velocity $\&$ Acceleration Vectors at $t=0$

[karush]

$\tiny{243.13.0113}$

$\textsf{The vector $r(t)$ is the position vector of a particle at time $t$.}]$
$\textsf{Find the angle between the velocity and the acceleration vectors at time $t=0$}\\$
\begin{align*} \displaystyle
r_{13}(t)&=sin^{-1}(4t)\large\textbf{i}+\ln(7t^2+1)\large\textbf{j}+\sqrt{8t^2+1}\large\textbf{k}\\
v_{13}(t)&=\frac{4}{\sqrt{1 - 16 t^2}}\large\textbf{i}
+ \frac{14 t}{(7t^2+1)}\large\textbf{j}
+ \frac{8 t}{\sqrt{8 t^2 + 1}}\large\textbf{k}\\
a_{13}(t)&=a_{13}(t)=\frac{64t}{(1- 16t^2)^{3/2}}\large\textbf{i}
+\frac{14-98t^2}{(7t^2+1)^2}\large\textbf{j}
+\frac{8}{(8t^2+1)^{3/2}}\large\textbf{k}\\
\textit{book answer}&=\color{red}{\frac{\pi}{2}}
\end{align*}

ok this looks kinda hefty
can we plug in the $t=0$
before the leap of faith into dot product or no?(drink)

 

[tkhunny]

Sometimes, a quick search will help you along your way...

https://www.freemathhelp.com/forum/...gle-Between-Velocity-and-Acceleration-Vectors

I think you have the right idea. Go without fear!

 

[karush]

Sometimes, a quick search will help you along your way...
https://www.freemathhelp.com/forum/...gle-Between-Velocity-and-Acceleration-Vectors
I think you have the right idea. Go without fear!

What... linked to the competition?

if $t=0$ then

\begin{align*} \displaystyle
v_{13}(0)&=4\large\textbf{i} +0\large\textbf{j} +0\large\textbf{k}\\
a_{13}(0)&=0\large\textbf{i} +14\large\textbf{j} +8\large\textbf{k}\\
\end{align*}

and then

\begin{align*}
\theta&=\cos^{-1}
\left[\frac{u\cdot v}{|u||v|} \right]
=\cos^{-1}
\left[\frac{(4,0,0)\cdot (0,14,8)}{|u||v|} \right]
=\cos^{-1}(0)=\frac{\pi}{2}
\end{align*}

suggestions?

 

[tkhunny]

What... linked to the competition?

Oops. My bad. I was on both sites, today, and forgot which one was current.

 

[I like Serena]

Oops. My bad. I was on both sites, today, and forgot which one was current.

Heh. As far as I'm concerned, I don't mind a bit of competition - as long as we end up above sites that are merely spamming! ;)
I'm not familiar with the mentioned site, but I definitely know that some other sites are full of spam and adds, which is something we do not allow or support.