Kinematics with constant velocity Problem set B

Find the bearing on which each object moves, to the nearest degree. The first component is east, the second north. Velocities are in m s⁻¹.

1. (a) A kayak: \(\mathbf{v} = \begin{pmatrix}8\\3\end{pmatrix}\)
2. (b) A rescue boat: \(\mathbf{v} = 5\mathbf{i} - 6\mathbf{j}\)

Find the speed of each object. Give each answer exactly and to 3 significant figures. Velocities are in m s⁻¹.

1. (a) A cyclist: \(\mathbf{v} = \begin{pmatrix}-6\\11\end{pmatrix}\)
2. (b) A rocket: \(\mathbf{v} = 2\mathbf{i} - 6\mathbf{j} + 9\mathbf{k}\)
3. (c) A bird: \(\mathbf{v} = \begin{pmatrix}7\\2\\-4\end{pmatrix}\)

Find the velocity vector of an object moving in the given direction at the given speed.

1. (a) Direction \(\begin{pmatrix}20\\21\end{pmatrix}\), speed 58 m s⁻¹
2. (b) Direction \(4\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}\), speed 18 m s⁻¹
3. (c) A diving robot starts at the point (5, −2, 1) and moves at 14 m s⁻¹ in the direction of \(3\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}\). Write down its position vector at time \(t\) seconds.

A model train moves with constant velocity \(\mathbf{v} = -\mathbf{i} + 3\mathbf{j} + 2.5\mathbf{k}\) m s⁻¹. At \(t = 0\) it is at the point (7, −1, −4) (distances in metres, time in seconds).

1. (a) Find the speed.
2. (b) Write down an equation for the position vector at time t.
3. (c) Does it pass through the point (5, 5, 2)? Give a reason.

A submarine moves so that its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}1\\-5\\2\end{pmatrix} + t\begin{pmatrix}2\\4\\4\end{pmatrix}\), where distances are in metres.

1. (a) Find the initial position of the object.
2. (b) Find the speed of the object.
3. (c) Find the distance of the object from the origin after 3 seconds.

Two birds have position vectors \(\mathbf{r}_1 = \begin{pmatrix}7\\1\\5\end{pmatrix} + t\begin{pmatrix}-2\\0.5\\1\end{pmatrix}\) and \(\mathbf{r}_2 = \begin{pmatrix}-1\\6\\2\end{pmatrix} + t\begin{pmatrix}1\\-1.5\\2\end{pmatrix}\) at time \(t\) seconds, where distances are in m. Find the distance between the birds when \(t = 4\).

Hiker R starts at \(9\mathbf{i} - 2\mathbf{j}\) km from a hut and walks with velocity \(-3\mathbf{i} + 4\mathbf{j}\) km h⁻¹. Hiker S starts at \(-6\mathbf{i} + 10\mathbf{j}\) km and walks with velocity \(2\mathbf{i} - \mathbf{j}\) km h⁻¹. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) point east and north.

1. (a) Find expressions for the position vectors of R and S after \(t\) hours.
2. (b) Find the distance between R and S after 2 hours.
3. (c) Find the time at which R is due north of S.

Two seagulls move in straight lines. Their position vectors at time \(t\) seconds are \(\mathbf{r}_1 = \begin{pmatrix}-10\\8\\-1\end{pmatrix} + t\begin{pmatrix}3\\-1\\2\end{pmatrix}\) and \(\mathbf{r}_2 = \begin{pmatrix}-8\\-2\\7\end{pmatrix} + t\begin{pmatrix}1\\2\\-1\end{pmatrix}\).

1. (a) Show that the paths cross, and find the point where they cross.
2. (b) Show that the objects do not collide.

Two robots move so that their position vectors at time \(t\) seconds are \(\mathbf{r}_1 = (1 + 2t)\mathbf{i} + (7 - 3t)\mathbf{j} + (-8 + 6t)\mathbf{k}\) and \(\mathbf{r}_2 = (-3 + 4t)\mathbf{i} + 1\mathbf{j} + (10 - 3t)\mathbf{k}\), where distances are in metres.

1. (a) Find the speed of each robot.
2. (b) Determine whether the robots meet.

Two joggers run across a park. Jogger P starts at \(7\mathbf{i}\) km from a gate and runs with velocity \(-\mathbf{i} - 2\mathbf{j}\) km h⁻¹. Jogger Q starts at \(\mathbf{i} - 6\mathbf{j}\) km and runs with velocity \(3\mathbf{i}\) km h⁻¹. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) point east and north. Time \(t\) is in hours.

1. (a) Show that the distance between P and Q at time \(t\) is \(d = \sqrt{20t^2 - 72t + 72}\).
2. (b) Use your calculator to find the least distance between them.
3. (c) Find the time at which Q is due east of P.

The position vectors of two paragliders at time \(t\) minutes are \(\mathbf{r}_1 = \begin{pmatrix}-3\\5\\1\end{pmatrix} + t\begin{pmatrix}2\\-1\\4\end{pmatrix}\) and \(\mathbf{r}_2 = \begin{pmatrix}6\\-4\\10\end{pmatrix} + t\begin{pmatrix}-1\\2\\1\end{pmatrix}\), with distances in m.

1. (a) Show that the paragliders would collide, and find the position vector of the point of collision.
2. (b) After 2 minutes, the second paraglider changes its velocity to \(\begin{pmatrix}0\\3\\1\end{pmatrix}\) to avoid the collision. Find the distance between the two paragliders at the time they would have collided.

Two moths move so that their position vectors at time \(t\) seconds are \(\mathbf{r}_1 = \begin{pmatrix}-2\\5\\2.5\end{pmatrix} + t\begin{pmatrix}1\\-2\\0.5\end{pmatrix}\) and \(\mathbf{r}_2 = \begin{pmatrix}4\\0.5\\1.2\end{pmatrix} + t\begin{pmatrix}-1\\-0.5\\0.4\end{pmatrix}\), in metres. The base vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the horizontal plane and \(\mathbf{k}\) points upwards. Call them the first and second moth.

1. (a) Show that there is a time when the first moth is vertically above the second moth, and find this time.
2. (b) Find the distance between them at that time.

Distances are in kilometres and time in hours. A hot-air balloon moves with constant velocity \(15\mathbf{i}\) km h⁻¹ and at \(t = 0\) it is at the origin. A drone is at the point (0, 0.9, −0.15) at \(t = 0\). It then moves with constant velocity in the direction of the vector \(20\mathbf{i} - 12\mathbf{j} + c\mathbf{k}\).

1. (a) Given that the drone reaches the hot-air balloon, find the value of \(c\).
2. (b) Find the speed of the drone.