IB Mathematics Applications & Interpretation Topic…

A survey drone is monitored from a fixed launch point. Its position relative to the launch point at time $t$ seconds is given by $r=\begin{pmatrix}9\\-12\\20\end{pmatrix}+t\begin{pmatrix}0.6\\-0.8\\2.4\end{pmatrix}$ . The components represent easting (m), northing (m), and height above the launch point (m).

(a) the speed of the drone.

(b) the distance of the drone from the launch point when $t=0$ .

(e) the time when the drone is $24$ m above the warehouse floor.

(f) the Euclidean distance between the drone and the charging pad at that moment.

(g) If the drone's velocity becomes $\begin{pmatrix}-3\\4\\a\end{pmatrix}\text{ m min}^{-1}$ after adjusting its descent, the constant .

A particle moves in a plane such that its position vector (in metres) at time $t$ seconds, for $t \ge 0$ , is given by $\mathbf{r} = \begin{pmatrix} t^2 \\ t^2 - 4t \end{pmatrix}$ .

(a) Find the velocity vector, $\mathbf{v}$ , of the particle in terms of $t$ .

(b) Find the position vector of the particle at the instant its speed is at a minimum.

An inspection drone in a warehouse has position coordinates $(x,y,z)$ , where $x$ and $y$ are the horizontal displacements from the centre of a charging pad on the warehouse floor and $z$ is the height above the floor. All distances are in metres.

At 14:30, the drone is at $(15,-20,36)$ . At 14:32, it is at $(9,-12,28)$ . Let $t$ be the time in minutes after 14:30.

Assume the drone continues to move in a straight line with constant velocity.

When it reaches a height of $24$ m, it keeps the same horizontal motion but changes its rate of descent so that it arrives exactly at $(0,0,0)$ .

(a) Formulate a vector equation for the position $\mathbf{r}$ of the drone as a function of $t$ .

(b) that the drone's path goes directly over the charging pad.

(d) the specific time it passes over the charging pad.

A remotely operated underwater vehicle moves with constant velocity. Its position relative to a fixed beacon at time $t$ seconds is given by $\mathbf{r}(t)=\begin{pmatrix}12-4t\\3+2t\\5-t\end{pmatrix}$ , where position is in metres.

(a) Find the initial position of the vehicle.

(b) the velocity vector of the vehicle.

A wildlife-monitoring drone $D$ flies over a flat nature reserve with constant velocity $v=\begin{pmatrix}3\\4\end{pmatrix}$ m/s. At time $t=0$ , the drone is at point $P(-15,10)$ relative to the origin $O$ , with coordinates in metres. A protected nesting site is fixed at $(25,30)$ .

(a) the position vector $\vec{OD}$ of the drone as a function of time $t$ .

(b) the value of $t$ at which the drone is closest to the nesting site.

(c) A no-fly alert is triggered if the drone passes within $18$ metres of the nesting site. if the alert will be triggered, providing a justification.

IB Mathematics Applications & Interpretation AHL 3.12—vector Applications to Kinematics Questionbank

IB Mathematics Applications & Interpretation AHL 3.12—vector Applications to Kinematics Questionbank

IB Mathematics Applications & Interpretation AHL 3.12—vector Applications to Kinematics Questionbank

IB Mathematics Applications & Interpretation AHL 3.12—vector Applications to Kinematics Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 3.12—vector Applications to Kinematics Questionbank