Kinematic Variables: Position, Velocity, and Acceleration as Vectors

A delivery drone is at the point with coordinates $(4, 3)$ on a map, with distances in meters. 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 meters. 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 meters, 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 center 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 meters.

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

A weather station measures the wind velocity as $\vec{v} = 12\,\vec{i} - 5\,\vec{j}\ \text{m/s}$, where the $x$ axis points east and the $y$ axis points north. The figure shows the wind velocity vector and its components.

1. a) Calculate the magnitude of the wind velocity.
2. b) Calculate the angle it makes with the positive $x$ axis and describe in words which way the wind is blowing.

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 meters, and, if so, state at what time it does so.

A trolley on a track moves according to $\vec{r}(t) = t^2\,\vec{i}$, in SI units. The $x$-$t$ graph of the trolley, showing two of those secants and the tangent line at $t = 1\ \text{s}$, is shown in the figure.

1. a) Calculate its average velocity between $t = 1\ \text{s}$ and $t = 3\ \text{s}$.
2. b) Repeat the calculation for the intervals $[1, 2]\ \text{s}$, $[1;\ 1.1]\ \text{s}$ and $[1;\ 1.01]\ \text{s}$.
3. c) Observe the sequence of values obtained, predict the value of the instantaneous velocity at $t = 1\ \text{s}$ and justify the reasoning that leads you to this value.

A ferry leaves port A, located at point $(2, 3)$, and arrives at port B, at point $(14, 15)$, with coordinates in kilometers. The crossing takes $2\ \text{h}$.

1. a) Calculate the displacement vector.
2. b) Calculate the average velocity vector in $\text{km/h}$ and its magnitude.
3. c) Express the magnitude of the average velocity in $\text{m/s}$, to one decimal place.

A rental e-scooter sets off from the origin and its position, in SI units, is $\vec{r}(t) = 3t\,\vec{i} + 2t\,\vec{j}$.

1. a) Calculate its position when $t = 10\ \text{s}$.
2. b) At what time does it pass through a point with $x = 90\ \text{m}$ coordinate? What is its full position at that moment?
3. c) Obtain the equation of the trajectory $y(x)$ by eliminating time, and deduce what type of curve it is.

A sensor in a laboratory records that a cart is at point $(2, 5)$ when $t_1 = 1\ \text{s}$ and at point $(14, 17)$ when $t_2 = 5\ \text{s}$, with positions in meters.

1. a) Calculate the displacement of the cart and the elapsed time interval.
2. b) Calculate the average velocity vector and its magnitude.
3. c) What angle does the average velocity make with the $x$ axis? Justify your answer using the components.

The position of a particle, in SI units, is $\vec{r}(t) = 3t\,\vec{i} + (t^2 + 1)\,\vec{j}$.

1. a) Calculate the positions at $t = 0$ and at $t = 4\ \text{s}$.
2. b) Calculate the displacement in that interval and its magnitude.
3. c) Calculate the average velocity vector in that interval and its magnitude.
4. d) The trajectory between both points is curved: deduce whether the average speed in that interval can equal the magnitude of the average velocity.

The position of a boat, in SI units, is $\vec{r}(t) = (2 + 3t)\,\vec{i} + 4t\,\vec{j}$.

1. a) What is its initial position?
2. b) Obtain the equation of the trajectory $y(x)$ and classify it.
3. c) Check whether it passes through the point $(8, 8)$, in meters, and state at what time.
4. d) Calculate the distance from the boat to the origin at that moment.

A person walks $30\ \text{m}$ to the east in $10\ \text{s}$ and then $40\ \text{m}$ to the north in another $10\ \text{s}$.

1. a) Calculate the total distance covered and the average speed.
2. b) Calculate the total displacement, its magnitude and the magnitude of the average velocity.
3. c) Compare the results of the two parts and state the general relationship between average speed and the magnitude of average velocity, indicating when they coincide.

A robotic lawnmower moves through a garden according to $\vec{r}(t) = 2t\,\vec{i} + (12 - t)\,\vec{j}$, in SI units.

1. a) Calculate its initial position.
2. b) Determine the time at which it crosses the angle bisector of the first quadrant, that is, when its coordinates $x$ and $y$ coincide, and its position at that moment.
3. c) Calculate its distance from the origin at that time.
4. d) Calculate the average velocity between $t = 0$ and that time.

Two surveillance drones move, in SI units, according to $\vec{r}_A(t) = 5t\,\vec{i} + 3t\,\vec{j}$ and $\vec{r}_B(t) = (60 - 5t)\,\vec{i} + 18\,\vec{j}$. The paths of both drones are shown in the figure.

1. a) Write the equations $x(t)$ and $y(t)$ for each drone.
2. b) Determine the time and position at which they meet.
3. c) Calculate the distance from the meeting point to the origin.
4. d) Deduce whether the paths of both drones could intersect at any other point.