Vector kinematic variables

A delivery drone is at the point with coordinates $(4, 3)$ on a map, with distances in metres. The figure shows the point and its position vector.

1. a) Write its position vector $\vec{r}$ in terms of the unit vectors $\vec{i}$ and $\vec{j}$.
2. b) Calculate the magnitude of $\vec{r}$ and explain what it represents physically.

An athlete completes exactly one lap of a track and returns to the starting point after covering $400\ \text{m}$.

1. a) What is their displacement?
2. b) What is the distance they covered?
3. c) Explain in your own words the difference between distance covered and displacement.

A robot in an automated warehouse moves from position $\vec{r}_1 = 2\,\vec{i} + \vec{j}$ to position $\vec{r}_2 = 8\,\vec{i} + 9\,\vec{j}$, with positions in metres. The figure shows both positions and the displacement between them.

1. a) Calculate the displacement vector $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$.
2. b) Calculate its magnitude.

A bicycle courier goes from the shop to a house and returns to the shop along the same route.

1. a) What is their total displacement?
2. b) What is their average velocity over the entire journey?
3. c) Is their average speed also zero? Justify your answer and explain the difference between average velocity and average speed.

A hiker leaves the refuge and covers $1,2\ \text{km}$ to the east and then $1,6\ \text{km}$ to the north, in a total time of $0,5\ \text{h}$. The diagram shows the hiker's route.

1. a) Calculate the magnitude of their total displacement.
2. b) Write their average velocity vector in $\text{km/h}$ and calculate its magnitude.
3. c) Calculate the angle that their displacement makes with the easterly direction.

A skater moves in a straight line, covering a displacement of $20\ \text{m}$ in the direction of the $x$ axis in $4\ \text{s}$.

1. a) Write her displacement vector in terms of the unit vector $\vec{i}$.
2. b) Calculate her average velocity $\vec{v}_m$ as a vector, remembering that $\vec{v}_m = \frac{\Delta\vec{r}}{\Delta t}$.

The displacement of an orienteer between two checkpoints is $\Delta\vec{r} = 30\,\vec{i} - 40\,\vec{j}$, in metres, and is completed in $10\ \text{s}$.

1. a) Calculate their average velocity vector.
2. b) Calculate the magnitude of this average velocity.

A cleaning robot in a shopping centre is at position $\vec{r}_1 = \vec{i} + \vec{j}$ when $t = 0$ and at position $\vec{r}_2 = 7\,\vec{i} + 9\,\vec{j}$ when $t = 2\ \text{s}$, with positions in metres.

1. a) Calculate the displacement vector and its magnitude.
2. b) Calculate the average velocity vector and its magnitude.

A motorcycle travelling in a straight line goes from $\vec{v}_1 = 5\,\vec{i}\ \text{m/s}$ to $\vec{v}_2 = 20\,\vec{i}\ \text{m/s}$ in $5\ \text{s}$. The initial and final velocity vectors are shown in the figure.

1. a) Calculate the change in velocity $\Delta\vec{v} = \vec{v}_2 - \vec{v}_1$.
2. b) Calculate its average acceleration $\vec{a}_m = \frac{\Delta\vec{v}}{\Delta t}$.
3. c) Explain what the result means: by how much does its velocity change every second?

An aircraft performing manoeuvres goes from $\vec{v}_1 = 2\,\vec{i} + 3\,\vec{j}\ \text{m/s}$ to $\vec{v}_2 = 8\,\vec{i} + 9\,\vec{j}\ \text{m/s}$ in $2\ \text{s}$. The figure shows both velocity vectors.

1. a) Calculate the change in velocity $\Delta\vec{v}$.
2. b) Calculate the average acceleration vector and its magnitude.

A car enters a roundabout at a velocity of $\vec{v}_1 = 12\,\vec{i}\ \text{m/s}$ and leaves it with $\vec{v}_2 = 12\,\vec{j}\ \text{m/s}$ after $3\ \text{s}$. The figure shows the path on the roundabout and both velocity vectors.

1. a) Check that the magnitude of the velocity has not changed.
2. b) Calculate the change in velocity $\Delta\vec{v}$ and its magnitude.
3. c) Calculate the average acceleration vector and its magnitude.
4. d) Explain why there is an acceleration if the speed has remained constant.

The position of a laboratory trolley is given, in SI units, by $\vec{r}(t) = 3t\,\vec{i} + 2\,\vec{j}$.

1. a) What is its initial position?
2. b) Calculate its position when $t = 4\ \text{s}$.
3. c) Explain why the trajectory is a horizontal straight line: notice what happens to the $y$ coordinate.

The position of a remote-controlled boat in a pond, in SI units, is $\vec{r}(t) = 2t\,\vec{i} + t^2\,\vec{j}$. The trajectory and both positions are shown in the figure.

1. a) Calculate the positions at times $t_1 = 1\ \text{s}$ and $t_2 = 3\ \text{s}$.
2. b) Calculate the displacement between these times and its magnitude.
3. c) Calculate the average velocity in this interval and its magnitude.

A particle moves according to $\vec{r}(t) = 2t\,\vec{i} + t^2\,\vec{j}$, in SI units.

1. a) Write the parametric equations $x(t)$ and $y(t)$.
2. b) Eliminate time and obtain the equation of the trajectory $y(x)$. What type of curve is it?
3. c) Check whether the particle passes through the point $(6, 9)$, in metres, and, if so, state at what time it does so.