Kinematics in vectors: notes and practice questions

(a) A deep-sea submersible is exploring an underwater trench. It is subjected to several forces, all measured in kN ($1$ kN $= 1000$ N).

Two main thrusters provide forces represented by the vectors $\begin{pmatrix} 45 \\ 10 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} 45 \\ -10 \\ 5 \end{pmatrix}$.

Two lateral thrusters exert forces of $\begin{pmatrix} 0 \\ 8 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ -8 \\ 0 \end{pmatrix}$.

The water resistance acting on the submersible is $\begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix}$ kN.

The buoyant force is $\begin{pmatrix} 0 \\ 0 \\ 120 \end{pmatrix}$ kN, and the weight of the submersible produces a force of $\begin{pmatrix} 0 \\ 0 \\ -100 \end{pmatrix}$ kN.

Find the resultant force acting on the submersible.

(b) Given that the mass of the submersible is $15\,000$ kg, find the acceleration of the submersible in ms$^{-2}$. The acceleration (ms$^{-2}$) of an object subject to a resultant force of $\mathbf{F}$ N is given by the formula $\mathbf{a} = \frac{\mathbf{F}}{m}$, where $m$ is the mass of the object in kilograms.

(c) The submersible is initially at the origin $(0,0,0)$ and has an initial velocity of $\begin{pmatrix} 10 \\ 0 \\ -5 \end{pmatrix}$ ms$^{-1}$.

(i) Find the displacement of the submersible at time $t$.

(ii) Find the displacement of the submersible at $t = 60$ s.

(a) A space probe, 'Voyager Alpha', is launched from a space station. The position of the probe at time $t$ days after launch is given by the vector

$\mathbf{r} = \begin{pmatrix} 10 \\ 20 \\ 5 \end{pmatrix} + t \begin{pmatrix} 80 \\ 40 \\ 10 \end{pmatrix}$

Distances are measured in thousands of kilometres.

Find the position vector of the probe $3$ days after launch.

(b) A second space probe, 'Explorer Beta', is launched from a different station. The position vector of this probe is given by the vector

$\mathbf{s} = \begin{pmatrix} -50 \\ -30 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 70 \\ 50 \\ 12 \end{pmatrix}$

Determine if the two flight paths intersect and, if so, state the point of intersection.

(c) The two probes were launched at the same time, so $\lambda = t$.

State, with a reason, whether the two probes actually collide.

(d) Calculate the distance between the two space stations (the initial launch points).

(e) Calculate the shortest distance that ever exists between the two probes and the time when this occurs. Assume $t \ge 0$ days.

(a) The position of a reconnaissance drone, relative to a control tower, is given by the vector equation $r = \begin{pmatrix} 7 \\ 2 \end{pmatrix} + t \begin{pmatrix} -3 \\ 4 \end{pmatrix}$, where $r$ is the position vector in metres and $t$ is the time in minutes.

Write down the position vector of the drone when $t = 0$ and when $t = 1$.

(b) Calculate the speed of the drone.

(c) Find an expression for the distance of the drone from the origin at time $t$.

(d) Hence find the minimum distance of the drone from the origin and the time at which it occurs.

A drone A takes off from a control tower at 10:00. It flies north-east at a horizontal speed of $70\sqrt{2} \text{ kmh}^{-1}$ and climbs at a rate of $3 \text{ kmh}^{-1}$. At 10:00, it is at a height of $5 \text{ km}$ directly above the control tower.

Find an expression for the displacement of drone A from the control tower at time $t$ hours after 10:00. Assume the control tower is at the origin (0,0,0) and the positive y-axis points North, and the positive x-axis points East.

At 10:30, a second drone B is $12 \text{ km}$ directly above the control tower. It flies on a bearing of $300^\circ$ at a horizontal speed of $80 \text{ kmh}^{-1}$ and descends at a rate of $2 \text{ kmh}^{-1}$.

Find an expression for the displacement of drone B from the control tower $t$ hours after 10:00.

Find the distance the two drones are apart when they have the same height.

(a) The position vector of Drone A at time $t$ seconds is given by $\boldsymbol{r}_A = 4 \cos(3t)\boldsymbol{i} + 5 \sin(3t)\boldsymbol{j}$, where displacement is measured in metres.

(i) Find an expression for the velocity of Drone A at time $t$.

(ii) Hence, find the speed of Drone A when $t = 1.2$ seconds.

(b) (i) Find an expression for the acceleration of Drone A at time $t$.

(ii) Show that the acceleration of Drone A is always directed towards the origin.

(c) The position vector of a second drone, Drone B, is given by $\boldsymbol{r}_B = -5 \sin(4t)\boldsymbol{i} + 4 \cos(4t)\boldsymbol{j}$.

For $0 \le t \le 10$, find the time when the two drones are closest to each other.

(d) At time $k$, where $0 < k < 1.5$, Drone B is moving parallel to Drone A.

(i) Find the value of $k$.

(ii) At time $k$, show that the two drones are moving in the opposite direction.

The position of a drone, D$_1$, $t$ seconds after leaving a control tower T, is given by

$\mathbf{r} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix} + t \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix}$, $t \ge 0$.

The units of distance are metres.

Write down the coordinates of the control tower T.

Four seconds after leaving T, D$_1$ is at point P.

Find the displacement vector $\vec{\text{TP}}$.

Find the distance $|\vec{\text{TP}}|$.

A second drone, D$_2$, leaves the control tower T at the same time as D$_1$. D$_2$ is moving in the direction of the vector $\begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$.

Find the angle between the initial flight paths of Drone 1 and Drone 2.

The drone D$_2$ has a speed of $10$ m s$^{-1}$.

Find the distance between Drone 1 and Drone 2 when $t = 4$ seconds.

A deep-sea research submersible, 'Nautilus', is being tracked relative to an underwater research station, 'Triton Base'. The coordinates $(x, y, z)$ represent the submersible's displacement in kilometres, where $x$ is east, $y$ is north, and $z$ is vertical displacement (positive upwards, so negative for depths below sea level).

At 10:00 AM, the submersible is detected at a position 60 km east and 24 km north of Triton Base, and at a depth of 15 km below sea level. Its velocity is given as $\begin{pmatrix} -120 \\ -48 \\ -10 \end{pmatrix}$ kmh$^{-1}$. Let $t$ be the length of time in hours from 10:00 AM.

Write down a vector equation for the displacement, $\vec{r}$, of the submersible in terms of $t$.

If the submersible continued to travel with the given velocity,

verify that it would pass directly over Triton Base (the point $(0,0,0)$);

state the depth of the submersible at this point;

find the time at which it would pass directly over Triton Base.

When the submersible is at a depth of 18 km below sea level, it continues to move horizontally on the same bearing but adjusts its vertical velocity so that it will dock precisely at Triton Base $(0,0,0)$.

Find the time at which the submersible is at a depth of 18 km below sea level.

Find the direct distance of the submersible from Triton Base at this point.

Given that the velocity of the submersible, after the adjustment of the vertical velocity, is $\begin{pmatrix} -120 \\ -48 \\ a \end{pmatrix}$ kmh$^{-1}$, find the value of $a$.

A robotic system is designed to launch small spherical 'seed pods' into a series of elevated collection bins. The robot is positioned at the origin, O, of a coordinate system on a horizontal platform. In this system, $x$ and $y$ represent the horizontal and vertical displacement from O, measured in metres.

Bin $B_1$ is the closest collection bin to the robot. The coordinates of the centre of bin $B_1$ are $(25, 1.5)$.

Each subsequent bin is $0.8$ m further from O horizontally and $0.4$ m higher than the bin in the row below it. Let bin $B_n$ be the bin in row $n$.

Write down the coordinates of the centre of bin $B_5$.

Find, in terms of $n$, the coordinates for the centre of bin $B_n$.

While in motion, a seed pod can be treated as a projectile.

Let $t$ be the time, in seconds, after a seed pod is launched.

At any time $t > 0$, the acceleration of the seed pod, in m s$^{-2}$, is given by the vector

$\begin{pmatrix} 0 \\ -9.8 \end{pmatrix}$

The initial velocity, in m s$^{-1}$, of the seed pod is given as $\begin{pmatrix} 24.5 \cos \theta \\ 24.5 \sin \theta \end{pmatrix}$, where $\theta$ is the angle to the horizontal at which the seed pod is launched and $0^\circ < \theta \le 90^\circ$.

Find an expression for the velocity, $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}$, at time $t$.

Hence show that when the seed pod is launched vertically, the time for it to reach its maximum height is $2.5$ seconds.

The displacement of the seed pod, $t$ seconds after it is launched, is given by the vector equation

$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 24.5(\cos \theta)t \\ 24.5(\sin \theta)t - 4.9t^2 \end{pmatrix}$

Using the given answer to part (b)(ii) or otherwise, find the maximum height reached by a seed pod when it is launched vertically.

If there were no bins to block its path, and the seed pod was launched at an angle $\theta$, show that the value of $x$ when it would hit the ground is given by the expression

$x = \frac{600.25}{9.8} (2 \sin \theta \cos \theta)$.

Hence find the maximum possible value for $x$ if there were no bins to block the path of the seed pod.

In order to calculate which bins can be reached by a seed pod, it is required to find the equation of the curve that forms the boundary of all the points that can be reached. This boundary is represented by a parabolic curve $y = ax^2 + bx + c$, with its vertex $V$ on the $y$-axis.

Using your answers to parts (c) and (d)(ii), or otherwise, find the value of $c$.

Find the value of $b$.

Find the value of $a$.

A technician is considering placing a special sensor in bin $B_{30}$.

Show that it is not possible for a seed pod to ever reach bin $B_{30}$.

Two drones, Alpha and Beta, are flying in a designated airspace. Their positions at time $t$ hours, $0 \le t < 15$, are given by the position vectors $r_A = \begin{pmatrix} 10 \\ 5 \\ 2 \end{pmatrix} + t \begin{pmatrix} -1 \\ -0.5 \\ 0.2 \end{pmatrix}$ and $r_B = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 0.6 \\ 0.3 \\ 0.4 \end{pmatrix}$ respectively, relative to a control tower (all lengths are in kilometres).

Show that the two drones would collide at a point P and write down the coordinates of P.

To avoid a collision, Drone Beta adjusts its velocity so that its position vector is now given by $r_B = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 0.3 \\ 0.15 \\ 0.2 \end{pmatrix}$.

Find the value of $t$ when Drone Beta, with its adjusted path, passes through point P.

Find the value of $t$ when the two drones (Drone Alpha and the adjusted Drone Beta) are closest together.

Find the distance between the two drones at this time.