Graphs of motion
A story goes: "You leave home on foot, stop at a traffic light and then carry on at the same pace." That story can be drawn without writing a single word: with a graph. Learning to read it is just as important as knowing how to do the calculations.
The position-time graph
Look at the first graph (position $x(t)$): it is a straight line. In the second (velocity $v(t)$) it is a horizontal line, because the speed does not change.
On a position-time graph ($x$-$t$ graph), which quantity is plotted on the horizontal axis and which on the vertical axis? In what units are they usually measured?
On a position-time graph ($x$-$t$ graph), which quantity is plotted on the horizontal axis and which on the vertical axis? In what units are they usually measured?
The $x$-$t$ graph of a cyclist is a horizontal line for $20\ \text{s}$, as shown in the figure. What is the cyclist doing during that time? What is their speed during…
The $x$-$t$ graph of a cyclist is a horizontal line for $20\ \text{s}$, as shown in the figure.
What is the cyclist doing during that time? What is their speed during this section?
The gradient gives the velocity; its absolute value, the speed
Two athletes set off from the finish line at the same time: after $5\ \text{s}$, one reaches the $20\ \text{m}$ mark and the other the $40\ \text{m}$ mark. If we draw their positions against time on the same axes, how can you tell at a glance which of the two is faster?
The $x$-$t$ graph of a runner is a straight line that starts at the origin and passes through the point marked in the figure ($t = 5\ \text{s}$, $x = 20\ \text{m}$). a)…
The $x$-$t$ graph of a runner is a straight line that starts at the origin and passes through the point marked in the figure ($t = 5\ \text{s}$, $x = 20\ \text{m}$).
- a) Where is the runner at $10\ \text{s}$?
- b) What is his speed?
On the same $x$-$t$ graph, the line for object A is steeper than that for object B, as shown in the figure. Which of the two is travelling faster? What relationship is…
On the same $x$-$t$ graph, the line for object A is steeper than that for object B, as shown in the figure.
Which of the two is travelling faster? What relationship is there between the steepness (the gradient) of the line and the speed?
On the same $x$-$t$ graph, object A goes from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $40\ \text{m}$) and object B goes from ($0\ \text{s}$, $0\ \text{m}$) to…
On the same $x$-$t$ graph, object A goes from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $40\ \text{m}$) and object B goes from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $20\ \text{m}$), as shown in the figure.
- a) Calculate the speed of each one.
- b) Which one is faster?
- c) What distance separates them at $10\ \text{s}$?
Sections: stationary, in URM or coming back
The $x$-$t$ graph of an object goes up to $x = 30\ \text{m}$ and then goes down to $x = 0\ \text{m}$, as shown in the figure. Explain in your own words what the object…
The $x$-$t$ graph of an object goes up to $x = 30\ \text{m}$ and then goes down to $x = 0\ \text{m}$, as shown in the figure.
Explain in your own words what the object has done.
The $x$-$t$ graph of an object has two sections: a straight line from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $30\ \text{m}$) and, from there, a horizontal…
The $x$-$t$ graph of an object has two sections: a straight line from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $30\ \text{m}$) and, from there, a horizontal section to ($20\ \text{s}$, $30\ \text{m}$), as shown in the figure.
- a) What is its speed in the first section?
- b) What does it do in the second section? How long does it last?
A student draws a $x$-$t$ graph with a section that goes "backwards" in time: according to their drawing, the object is at $x = 30\ \text{m}$ at $10\ \text{s}$ and at…
A student draws a $x$-$t$ graph with a section that goes "backwards" in time: according to their drawing, the object is at $x = 30\ \text{m}$ at $10\ \text{s}$ and at $x = 40\ \text{m}$ at $8\ \text{s}$, as shown in the figure.
Explain why that graph is impossible and what mistake the student made.
You can do it on paper and send a photo with your answer.
Average speed from the graph
The $x$-$t$ graph of an object rises from $x = 0\ \text{m}$ to $x = 60\ \text{m}$ and then falls to $x = 20\ \text{m}$, as shown in the figure. a) What total distance…
The $x$-$t$ graph of an object rises from $x = 0\ \text{m}$ to $x = 60\ \text{m}$ and then falls to $x = 20\ \text{m}$, as shown in the figure.
- a) What total distance has it covered?
- b) What has its displacement been?
- c) Do the two values match? Explain why.
The $x$-$t$ graph of a delivery driver has three sections: from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $60\ \text{m}$); horizontal up to ($20\ \text{s}$,…
The $x$-$t$ graph of a delivery driver has three sections: from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $60\ \text{m}$); horizontal up to ($20\ \text{s}$, $60\ \text{m}$); and from ($20\ \text{s}$, $60\ \text{m}$) to ($50\ \text{s}$, $0\ \text{m}$), as shown in the figure.
- a) Calculate the speed of each section.
- b) What is the delivery driver doing in each section? Make up a short story that fits the graph.
- c) Calculate the total distance covered and the displacement.
- d) Calculate the average speed for the whole journey.
The $x$-$t$ graph of a mountain hike is: from ($0\ \text{min}$, $0\ \text{m}$) to ($30\ \text{min}$, $1200\ \text{m}$); horizontal up to ($40\ \text{min}$,…
The $x$-$t$ graph of a mountain hike is: from ($0\ \text{min}$, $0\ \text{m}$) to ($30\ \text{min}$, $1200\ \text{m}$); horizontal up to ($40\ \text{min}$, $1200\ \text{m}$); and back from ($40\ \text{min}$, $1200\ \text{m}$) to ($80\ \text{min}$, $0\ \text{m}$), as shown in the figure.
- a) Calculate the speed for the journey there and the journey back, in $\text{m/min}$.
- b) How long were they resting for?
- c) Calculate the average speed for the whole excursion.
- d) The speed on the way back is lower than on the way there. Suggest a reasonable explanation.
Back to the beginning: The walk in the story is drawn as a straight line sloping upwards (walking), a flat horizontal section (waiting at the traffic light) and another straight line sloping upwards with the same steepness as the first (carrying on at the same pace).