Answer
$2ti+2+\dfrac{1}{t}k$, $2i-\dfrac{1}{t^2}k$ , $2t+\dfrac{1}{t}$
Work Step by Step
Given: $r(t)=t^2i+2t j+\ln t k$
Our aim is to calculate the velocity vector, acceleration vector and speed.
In order to calculate the all above terms we will use formulas, such as:
$v(t)=r'(t)$ and $a(t)=v'(t)$ and speed is the magnitude of the velocity vector, that is $s(t)=|v(t)|$.
Now,
$v(t)=r'(t)=2ti+2+\frac{1}{t}k$
$a(t)=v'(t)=2i-\frac{1}{t^2}k$
$s(t)=|v(t)|=\sqrt {(2t)^2+(2)^2+(\frac{1}{t})^2}=2t+\frac{1}{t}$
Hence, the required answers are:
$2ti+2+\dfrac{1}{t}k$, $2i-\dfrac{1}{t^2}k$ , $2t+\dfrac{1}{t}$
