[填空题]
Adriano is riding a skateboard in a pa cm.* 5bsi5wqiam2d j3rking lot. His position vecto q).a,kp5 yeutda q,(nr from a fixed or yd ( qtu,pe.ka5,a)qnigin $\mathrm{O}$ 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.
1. Find the velocity vector at time t .
2. 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) $.
3. Find the value of a and the value of b . a = b =
4. 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. ≈ $ms^{-1}$
5. Find $|\mathrm{OP}| $. ≈ m
