Question
Adriano is riding a skateboard in a parking lot. His position vector from a fixed origin \(\mathrm{O}\) at time \(t\) seconds is modelled by
\[
\left(\begin{array}{l}
x \\
y
\end{array}\right)=\left(\begin{array}{l}
a \ln (t+b) \cos t \\
a \ln (t+b) \sin t
\end{array}\right)
\]
where \(a\) and \(b\) are non-zero constants to be determined. All distances are in metres.
(a) Find the velocity vector at time \(t\).
(b) Given that \(a>0\), show that the magnitude of the velocity vector at time \(t\) is given by \(a \sqrt{\frac{1}{(t+b)^2}+(\ln (t+b))^2}\).
At time \(t=0\), the velocity vector is \(\left(\begin{array}{c}2 \\ 2.773\end{array}\right)\).
(c) Find the value of \(a\) and the value of \(b\).
(d) Find the magnitude of the velocity vector when \(t=3\).
At point \(\mathrm{P}\), Adriano is riding parallel to the \(y\)-axis for the first time.
(e) Find \(|\mathrm{OP}|\).