Kinematic variables: displacement, velocity and acceleration
In Junior Cycle Science you described motion using numbers: a displacement $s$, a velocity $v$, a sign that indicated the direction. On this page you make that precise for motion in a straight line: a Luas on the track, a sliotar struck along the ground, a car on the M50. You will separate distance from displacement and speed from velocity, and define acceleration with its sign. Motion in a plane (position vectors, unit vectors, parametric equations) is an extension, marked in the boxes below.
Displacement, velocity and acceleration in a straight line
Displacement $s$ is the change in position, final minus initial, with a sign. Distance travelled is the total length of the path and is never negative. Average speed is distance travelled divided by time; average velocity is $v = \Delta s/\Delta t$. Acceleration is $a = \Delta v/\Delta t$, in $\mathrm{m\,s^{-2}}$, and its sign shows the direction of the change in velocity.
In a straight line, write displacement, velocity and acceleration with a sign: for example, a displacement of $+20\ \text{m}$ means 20 m in the positive direction. In the extension sections below, the same quantities are written as vectors using the unit vectors $\vec{i}$ and $\vec{j}$, so that $20\,\vec{i}\ \text{m}$ is just $+20\ \text{m}$ along the $x$ axis.
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An athlete completes exactly one lap of a track and returns to the starting point after covering $400\ \text{m}$. a) What is their displacement? b) What is the distance…
An athlete completes exactly one lap of a track and returns to the starting point after covering $400\ \text{m}$.
a) What is their displacement?
b) What is the distance they covered?
c) Explain in your own words the difference between distance covered and displacement.
A bicycle courier goes from the shop to a house and returns to the shop along the same route. a) What is their total displacement? b) What is their average velocity over…
A bicycle courier goes from the shop to a house and returns to the shop along the same route.
a) What is their total displacement?
b) What is their average velocity over the entire journey?
c) Is their average speed also zero? Justify your answer and explain the difference between average velocity and average speed.
On a displacement–time graph, the average velocity between two times is the gradient of the line (secant) joining the two points. The velocity at one instant is the gradient of the tangent to the curve at that point. As the time interval gets smaller, the secant gradient approaches the tangent gradient.
The exercise below writes the position as a vector, which is an extension; only the gradient idea is needed at this level.
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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…
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.
a) Calculate its average velocity between $t = 1\ \text{s}$ and $t = 3\ \text{s}$.
b) Repeat the calculation for the intervals $[1, 2]\ \text{s}$, $[1;\ 1.1]\ \text{s}$ and $[1;\ 1.01]\ \text{s}$.
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.
If a drone lands in Phoenix Park and you want to tell a classmate where it is, it is not enough to say that it is $20\ \text{m}$ away: you need to specify how many metres to go east and how many north. How do we represent, in a single mathematical entity, the distance and direction of a point relative to the observer?
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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. a) Write its position…
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.
a) Write its position vector $\vec{r}$ in terms of the unit vectors $\vec{i}$ and $\vec{j}$.
b) Calculate the magnitude of $\vec{r}$ and explain what it represents physically.
In a straight line, displacement is simply $s = s_{final} - s_{initial}$, with a sign. Distance travelled is the total length of the path and is never negative. If a cyclist goes from $s_1 = 15\ \text{m}$ to $s_2 = 5\ \text{m}$, the displacement is $5 - 15 = -10\ \text{m}$ and the distance travelled is $10\ \text{m}$.
A delivery cyclist in Cork rides $3\ \text{km}$ along winding streets to deliver an order to a house next to the base, which is only $30\ \text{m}$ away in a straight line. How do we distinguish mathematically the total path travelled from the net change in position between departure and arrival?
The exercises below use displacement vectors in a plane, so they are extension exercises.
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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…
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.
a) Calculate the displacement vector $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$.
A skater moves in a straight line, covering a displacement of $20\ \text{m}$ in the direction of the $x$ axis in $4\ \text{s}$. a) Write her displacement vector in terms…
A skater moves in a straight line, covering a displacement of $20\ \text{m}$ in the direction of the $x$ axis in $4\ \text{s}$.
a) Write her displacement vector in terms of the unit vector $\vec{i}$.
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 $10\ \text{s}$. a) Calculate…
The displacement of an orienteer between two checkpoints is $\Delta\vec{r} = 30\,\vec{i} - 40\,\vec{j}$, in metres, and is completed in $10\ \text{s}$.
a) Calculate their average velocity vector.
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…
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.
a) Calculate the displacement vector and its magnitude.
b) Calculate the average velocity vector and its magnitude.
You are standing on the Luas at a constant $50\ \mathrm{km\,h^{-1}}$; when the train goes into a sharp bend without braking or accelerating, you feel pushed outwards because your body tends to continue in a straight line while the train turns, and you have to hold on. Why do you feel the effects of an acceleration if the speedometer has not changed at all?
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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…
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.
a) Calculate the change in velocity $\Delta\vec{v} = \vec{v}_2 - \vec{v}_1$.
b) Calculate its average acceleration $\vec{a}_m = \frac{\Delta\vec{v}}{\Delta t}$.
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…
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.
a) Calculate the change in velocity $\Delta\vec{v}$.
b) Calculate the average acceleration vector and its magnitude.
The position of a laboratory trolley is given, in SI units, by $\vec{r}(t) = 3t\,\vec{i} + 2\,\vec{j}$. a) What is its initial position? b) Calculate its position when…
The position of a laboratory trolley is given, in SI units, by $\vec{r}(t) = 3t\,\vec{i} + 2\,\vec{j}$.
a) What is its initial position?
b) Calculate its position when $t = 4\ \text{s}$.
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.…
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.
a) Calculate the positions at times $t_1 = 1\ \text{s}$ and $t_2 = 3\ \text{s}$.
b) Calculate the displacement between these times and its magnitude.
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. a) Write the parametric equations $x(t)$ and $y(t)$. b) Eliminate time and obtain…
A particle moves according to $\vec{r}(t) = 2t\,\vec{i} + t^2\,\vec{j}$, in SI units.
a) Write the parametric equations $x(t)$ and $y(t)$.
b) Eliminate time and obtain the equation of the trajectory $y(x)$. What type of curve is it?
c) Check whether the particle passes through the point $(6, 9)$, in metres, and, if so, state at what time it does so.