Interpreting x-t and v-t graphs
A story goes: "you leave home at a steady pace, speed up when you realise you are late, and slow down at the end". That story can be drawn without writing a single word, using a graph. And from a graph you can tell at a glance what type of motion there is in each stage: uniform motion, uniformly accelerated motion or rest.
The x-t graph: position and gradient
We measure the position of a ball in a corridor: at $0\ \text{s}$ it is at $0\ \text{m}$, at $2\ \text{s}$ at $4\ \text{m}$ and at $4\ \text{s}$ at $8\ \text{m}$; how can we see at a glance whether it is moving fast or slowly without relying on a table of numbers?
In the first graph ($x(t)$) look at the slope of the line; in the second ($v(t)$) check that it is a horizontal line, because the velocity does not change.
In a position-time (x-t) graph, the line of a moving object is horizontal for $10\ \text{s}$, as shown in the figure. What is the moving object doing during that time?…
In a position-time (x-t) graph, the line of a moving object is horizontal for $10\ \text{s}$, as shown in the figure.
What is the moving object doing during that time? Justify your answer.
On the same x-t graph, Ana and Luis appear as two straight lines, and Ana's line rises more steeply than Luis's, as shown in the figure. Who is travelling faster? What…
On the same x-t graph, Ana and Luis appear as two straight lines, and Ana's line rises more steeply than Luis's, as shown in the figure.
Who is travelling faster? What physical quantity does the gradient of an x-t graph give us?
The x-t graph of a walker is a straight line passing through the points $(0\ \text{s},\ 0\ \text{m})$ and $(10\ \text{s},\ 50\ \text{m})$, as shown in the figure.…
The x-t graph of a walker is a straight line passing through the points $(0\ \text{s},\ 0\ \text{m})$ and $(10\ \text{s},\ 50\ \text{m})$, as shown in the figure.
Calculate their speed.
The v-t graph: gradient = acceleration
The speedometer of a motorbike reads $0\ \text{m/s}$ at the start, rises to $10\ \text{m/s}$ after $2\ \text{s}$ and reaches $20\ \text{m/s}$ after $4\ \text{s}$; how can we draw how quickly that velocity changes?
Notice how, since $a \neq 0$, the $v(t)$ graph is no longer horizontal: it is a sloping line, and the $x(t)$ graph curves.
The velocity-time (v-t) graph of a cruising aircraft is a horizontal line at $v = 250\ \text{m/s}$ for $10\ \text{s}$, as shown in the figure. What type of motion does…
The velocity-time (v-t) graph of a cruising aircraft is a horizontal line at $v = 250\ \text{m/s}$ for $10\ \text{s}$, as shown in the figure.
What type of motion does it have in this section? What distance does it travel in those $10\ \text{s}$?
The v-t graph of a motorbike is a straight line passing through the points $(0\ \text{s},\ 0\ \text{m/s})$ and $(4\ \text{s},\ 12\ \text{m/s})$, as shown in the figure.…
The v-t graph of a motorbike is a straight line passing through the points $(0\ \text{s},\ 0\ \text{m/s})$ and $(4\ \text{s},\ 12\ \text{m/s})$, as shown in the figure.
Calculate its acceleration.
The v-t graph of a cyclist is a straight line going from $(0\ \text{s},\ 12\ \text{m/s})$ to $(6\ \text{s},\ 0\ \text{m/s})$, as shown in the figure. a) Calculate their…
The v-t graph of a cyclist is a straight line going from $(0\ \text{s},\ 12\ \text{m/s})$ to $(6\ \text{s},\ 0\ \text{m/s})$, as shown in the figure.
- a) Calculate their acceleration.
- b) What does the sign of this acceleration mean?
- c) What would it mean if the line continued below the axis, with negative velocities?
Signed area: displacement; absolute areas: distance
If $v<0$, the area contributes a negative displacement. For the distance, add up the absolute values of the areas, separating the stages in which the sign of $v$ changes. For example, $v=4\ \text{m/s}$ for $3\ \text{s}$ and $v=-2\ \text{m/s}$ for $2\ \text{s}$ give $\Delta x=12-4=8\ \text{m}$, but $d=12+4=16\ \text{m}$. In the following examples the velocities are positive, which is why distance and displacement coincide.
The v-t graph of a tram is horizontal at $10\ \text{m/s}$ between $t = 0\ \text{s}$ and $t = 20\ \text{s}$, and horizontal at $5\ \text{m/s}$ between $t = 20\ \text{s}$…
The v-t graph of a tram is horizontal at $10\ \text{m/s}$ between $t = 0\ \text{s}$ and $t = 20\ \text{s}$, and horizontal at $5\ \text{m/s}$ between $t = 20\ \text{s}$ and $t = 40\ \text{s}$, as shown in the figure.
- a) What distance does it travel in the first section? And in the second?
- b) What total distance does it travel in $40\ \text{s}$?
- c) What is its average speed over the entire journey?
The x-t graph of a school bus journey has these straight sections, as shown in the figure: from $(0\ \text{min},\ 0\ \text{km})$ to $(10\ \text{min},\ 6\ \text{km})$;…
The x-t graph of a school bus journey has these straight sections, as shown in the figure: from $(0\ \text{min},\ 0\ \text{km})$ to $(10\ \text{min},\ 6\ \text{km})$; horizontal up to $t = 15\ \text{min}$, while picking up students; from $(15\ \text{min},\ 6\ \text{km})$ to $(35\ \text{min},\ 16\ \text{km})$; and finally from $(35\ \text{min},\ 16\ \text{km})$ to $(55\ \text{min},\ 0\ \text{km})$.
- a) Calculate the speed of each section, in $\text{km/min}$.
- b) What does the last section mean, whose gradient is negative?
- c) In which section is the bus travelling fastest? Justify it by comparing the gradients.
Combined stages and curves
Match each description of an x-t graph with the corresponding motion and justify your choices: a) A horizontal straight line. b) A sloped straight line rising upwards.…
Match each description of an x-t graph with the corresponding motion and justify your choices:
- a) A horizontal straight line.
- b) A sloped straight line rising upwards.
- c) A curve getting progressively steeper.
Motions: it is stationary; it moves at a constant speed; it moves faster and faster.
The x-t graph of a skater has two straight sections: the first goes from $(0\ \text{s},\ 0\ \text{m})$ to $(10\ \text{s},\ 100\ \text{m})$ and the second from…
The x-t graph of a skater has two straight sections: the first goes from $(0\ \text{s},\ 0\ \text{m})$ to $(10\ \text{s},\ 100\ \text{m})$ and the second from $(10\ \text{s},\ 100\ \text{m})$ to $(30\ \text{s},\ 200\ \text{m})$, as shown in the figure.
Calculate their speed in each section and state in which one they are going faster.
The x-t graph of a bike ride has three straight sections, as shown in the figure: from $(0\ \text{min},\ 0\ \text{km})$ to $(20\ \text{min},\ 10\ \text{km})$; horizontal…
The x-t graph of a bike ride has three straight sections, as shown in the figure: from $(0\ \text{min},\ 0\ \text{km})$ to $(20\ \text{min},\ 10\ \text{km})$; horizontal from there to $t = 30\ \text{min}$; and from $(30\ \text{min},\ 10\ \text{km})$ to $(60\ \text{min},\ 16\ \text{km})$.
- a) Calculate the speed of each section, in $\text{km/min}$.
- b) What happens in the middle section? What do you think the cyclists did during that time?
The x-t graph of a car starting off is a curve that starts with a gentle slope and becomes progressively steeper, as shown in the figure. a) Does its speed increase,…
The x-t graph of a car starting off is a curve that starts with a gentle slope and becomes progressively steeper, as shown in the figure.
- a) Does its speed increase, decrease or stay constant? Reason your answer based on how the gradient changes.
- b) What would the curve look like if, instead, the car braked to a stop? Describe it in words.
Two moving objects on the same graph
On the same x-t graph, Carla's line starts at $(0\ \text{s},\ 30\ \text{m})$ and passes through $(10\ \text{s},\ 60\ \text{m})$, and Dani's line starts at…
On the same x-t graph, Carla's line starts at $(0\ \text{s},\ 30\ \text{m})$ and passes through $(10\ \text{s},\ 60\ \text{m})$, and Dani's line starts at $(0\ \text{s},\ 0\ \text{m})$ and passes through $(10\ \text{s},\ 60\ \text{m})$, as shown in the figure.
- a) Calculate each person's speed.
- b) Who started ahead? Who runs faster?
- c) Do they cross paths at any point? Where can this be seen on the graph?
On an x-t graph, the line of one train passes through $(0\ \text{s},\ 0\ \text{m})$ and $(20\ \text{s},\ 400\ \text{m})$, and that of another train, coming from the…
On an x-t graph, the line of one train passes through $(0\ \text{s},\ 0\ \text{m})$ and $(20\ \text{s},\ 400\ \text{m})$, and that of another train, coming from the opposite direction, passes through $(0\ \text{s},\ 800\ \text{m})$ and $(40\ \text{s},\ 0\ \text{m})$, as shown in the figure.
- a) Calculate the velocity of each train, including its sign.
- b) The two lines intersect at a point: explain what this point means.
- c) Calculate the time and position at which the trains pass each other.
A classmate draws the x-t graph of a runner with these two sections, as shown in the figure: first a straight line from $(0\ \text{s},\ 0\ \text{m})$ to…
A classmate draws the x-t graph of a runner with these two sections, as shown in the figure: first a straight line from $(0\ \text{s},\ 0\ \text{m})$ to $(10\ \text{s},\ 40\ \text{m})$ and then a straight line going from $(10\ \text{s},\ 40\ \text{m})$ to $(5\ \text{s},\ 20\ \text{m})$.
- a) Calculate the speed of the first section.
- b) Explain why the second section is impossible in reality: notice how time progresses.
- c) What should a section look like where the runner actually returns towards the starting point? Describe it and give a numerical example.
You can do it on paper and send a photo with your answer.
Back to the beginning: On a $v$-$t$ graph, your story corresponds to a horizontal line (uniform motion at a steady pace), a stage sloping upwards (you speed up with uniformly accelerated motion) and a stage sloping downwards to the time axis (you slow down until you stop).