Constant velocity: speed, position and closest approach Problem set B

A cyclist moves with constant velocity \(\mathbf{v} = \begin{pmatrix} 3 \\ -8 \end{pmatrix}\) m s\(^{-1}\), where \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) points east and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) points north.

1. (a) Find the direction of motion as a bearing.
2. (b) Find the speed.
3. (c) A second object has velocity \(\begin{pmatrix} -6 \\ 5 \end{pmatrix}\) m s\(^{-1}\). Find its direction of motion as a bearing.

A shark moves with constant speed in a fixed direction. Distances are in metres and time in seconds.

1. (a) The direction is \(\begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}\) and the speed is 21 m s\(^{-1}\). Find the velocity vector.
2. (b) The direction is \(\begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix}\) and the speed is 9 m s\(^{-1}\). Find the velocity vector, giving each component to 3 significant figures.

An object moves with constant velocity \(\mathbf{v} = \begin{pmatrix} 2 \\ -3 \\ -1 \end{pmatrix}\) m s\(^{-1}\). At \(t = 0\) its position vector is \(\mathbf{r}_0 = \begin{pmatrix} -5 \\ 1 \\ 8 \end{pmatrix}\) m, where \(t\) is in seconds.

1. (a) Write down an equation for the position vector of the object at time \(t\).
2. (b) Find the distance of the object from the origin when \(t = 3\).

Two submarines move with constant velocities. Their position vectors (in km) at time \(t\) hours are given.

1. (a) \(\mathbf{r}_A = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix} + t\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}\) and \(\mathbf{r}_B = \begin{pmatrix} 4 \\ -5 \\ 11 \end{pmatrix} + t\begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). Show that \(A\) and \(B\) do not collide.
2. (b) \(\mathbf{r}_C = \begin{pmatrix} -5 \\ 8 \\ -4 \end{pmatrix} + t\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r}_D = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + t\begin{pmatrix} -1 \\ 3 \\ 0 \end{pmatrix}\). Show that \(C\) and \(D\) collide, and find where.

Two divers have position vectors (in m) at time \(t\) minutes: \(\mathbf{r}_A = \begin{pmatrix} 0 \\ 6 \\ 5 \end{pmatrix} + t\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\) and \(\mathbf{r}_B = \begin{pmatrix} 9 \\ -3 \\ 1 \end{pmatrix} + t\begin{pmatrix} -1 \\ 3 \\ 0 \end{pmatrix}\).

1. (a) Find \(\overrightarrow{AB}\) in terms of \(t\).
2. (b) Use your calculator to find the time at which they are closest together, and their distance apart at that time.

A search robot starts at the point with position vector \(\begin{pmatrix} -20 \\ 35 \\ -5 \end{pmatrix}\) m, relative to the control point O, and moves in the direction \(\begin{pmatrix} -2 \\ 4 \\ 1 \end{pmatrix}\) at a constant speed of 6 m s\(^{-1}\).

1. (a) Find its velocity vector, giving each component to 3 significant figures.
2. (b) Write down its position vector at time \(t\) seconds.
3. (c) Find its distance from the control point O after 15 seconds.

Two marine drones have position vectors (in m) at time \(t\) seconds: \(\mathbf{r}_A = \begin{pmatrix} -4 \\ 2 \\ 6 \end{pmatrix} + t\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}\) and \(\mathbf{r}_B = \begin{pmatrix} 2 \\ -4 \\ -6 \end{pmatrix} + t\begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}\).

1. (a) Show that they do not collide.
2. (b) Find the time at which they are closest together, and the distance between them then.
3. (c) Safety rules require at least 3 m between them at all times. Is the rule kept? Give a reason.