Exercises: Motion graphs

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 an object is an upward straight line. Explain what this tells us about the object: is it moving or stationary? Is its speed constant or changing?

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 $x$-$t$ graph of a runner is a straight line that starts at the origin and passes through the point marked in the figure ($t = 5\ \text{s}$, $x = 20\ \text{m}$).

1. a) Where is the runner at $10\ \text{s}$?
2. 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 there between the steepness (the gradient) of the line and the speed?

The $x$-$t$ graph of a supermarket trolley is a straight line passing through the points ($0\ \text{s}$, $0\ \text{m}$) and ($10\ \text{s}$, $20\ \text{m}$). Calculate its speed.

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.

Fátima leaves her house, walks for a while, stops to talk to a friend and then carries on walking to the library. How many sections does the $x$-$t$ graph of her walk have? Describe the shape of each section.

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.

1. a) What is its speed in the first section?
2. b) What does it do in the second section? How long does it last?

The $x$-$t$ graph of a cyclist passes through the points ($0\ \text{s}$, $0\ \text{m}$), ($5\ \text{s}$, $20\ \text{m}$) and ($15\ \text{s}$, $40\ \text{m}$), as shown in the figure.

1. a) Calculate the speed for each section.
2. b) In which section is it travelling faster?
3. c) How can this be seen from the shape of the graph?

A story says: "A boy leaves home walking, stops at a traffic light and then carries on walking at the same pace." Which of these graphs corresponds to it? Justify your choice.

1. a) A straight line that rises continuously.
2. b) A rising section, a horizontal section and another section rising with the same gradient as the first.
3. c) A rising section, a horizontal section and a falling section.

The $x$-$t$ graph of a hot-air balloon is a straight line going from ($0\ \text{s}$, $100\ \text{m}$) to ($20\ \text{s}$, $0\ \text{m}$), as shown in the figure.

1. a) Is the balloon moving? In which direction?
2. b) What distance does it cover?
3. c) Calculate its 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 ($10\ \text{s}$, $20\ \text{m}$), as shown in the figure.

1. a) Calculate the speed of each one.
2. b) Which one is faster?
3. c) What distance separates them at $10\ \text{s}$?

A walker starts from position $x = 0\ \text{m}$ and travels at $2\ \text{m/s}$ for $10\ \text{s}$.

1. a) What shape does their $x$-$t$ graph have?
2. b) Which points does it pass through?
3. c) Draw it in your exercise book and describe it in words.

The $x$-$t$ graph of a bus rises from ($0\ \text{s}$, $0\ \text{m}$) to ($10\ \text{s}$, $40\ \text{m}$), is horizontal up to ($30\ \text{s}$, $40\ \text{m}$) and rises again up to ($40\ \text{s}$, $60\ \text{m}$), as shown in the figure.

1. a) How long was the bus stationary for?
2. b) Calculate the speed of the first section and the last section. Which is greater?

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.

1. a) What total distance has it covered?
2. b) What has its displacement been?
3. 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 ($20\ \text{s}$, $60\ \text{m}$); and from ($20\ \text{s}$, $60\ \text{m}$) to ($50\ \text{s}$, $0\ \text{m}$), as shown in the figure.

1. a) Calculate the speed of each section.
2. b) What is the delivery driver doing in each section? Make up a short story that fits the graph.
3. c) Calculate the total distance covered and the displacement.
4. d) Calculate the average speed for the whole journey.

Read this story: "María leaves home walking slowly towards the park; halfway there she realises she has forgotten her phone and runs back to get it; then she walks to the park, faster than at first."

1. a) How many sections does María's $x$-$t$ graph have?
2. b) Describe each section: does it rise or fall? Is it steep or gentle?
3. c) Compare the steepness of the first and third sections and justify your answer.

On a $x$-$t$ graph, object A sets off from $x = 0\ \text{m}$ at time $t = 0\ \text{s}$ travelling at $2\ \text{m/s}$. Object B sets off from the same point at $20\ \text{s}$ travelling at $4\ \text{m/s}$, as shown in the figure.

1. a) Where is A when B sets off?
2. b) The lines for the two objects intersect at a point. Explain what this point of intersection means.
3. c) Calculate at what time and at what position B catches up with object A.

The $x$-$t$ graph of a tram is a straight line passing through ($0\ \text{s}$, $0\ \text{m}$) and through ($20\ \text{s}$, $100\ \text{m}$), as shown in the figure.

1. a) What is its speed?
2. b) If it continues with the same MRU, what position will it be at after $60\ \text{s}$?
3. c) How long will it take to reach position $x = 400\ \text{m}$?

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.

The $x$-$t$ graph of a toy car has three consecutive sections: it rises $20\ \text{m}$ in $10\ \text{s}$, then another $60\ \text{m}$ in $10\ \text{s}$, and finally $20\ \text{m}$ in $20\ \text{s}$, as shown in the figure.

1. a) Calculate the speed of each section.
2. b) Which is the fastest section and which is the slowest?
3. c) How can fast and slow sections be identified on the graph without performing any calculations?

Read this story: "I leave my front door, cover $100\ \text{m}$ in $50\ \text{s}$, wait $25\ \text{s}$ for the bus to arrive and the bus takes me another $200\ \text{m}$ in $25\ \text{s}$."

1. a) Write down the key points of the $x$-$t$ graph, with time in seconds and position in metres.
2. b) Calculate the speed of each of the three sections.
3. c) In which section is the speed greatest? How would this be noticed on the graph?

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.

1. a) Calculate the speed for the journey there and the journey back, in $\text{m/min}$.
2. b) How long were they resting for?
3. c) Calculate the average speed for the whole excursion.
4. d) The speed on the way back is lower than on the way there. Suggest a reasonable explanation.

Unit challenge. Do you move with genuine uniform rectilinear motion (URM)? Become a researcher and analyse your own kinematics in the hallway at home to find out.

Phase 1. Experimental measurement: Mark a straight line on the floor with positions $0\text{ m}$, $2\text{ m}$, $4\text{ m}$, and $6\text{ m}$, recording an uncertainty of $\pm 0,02\text{ m}$. Ask someone to help you time (uncertainty of $\pm 0,2\text{ s}$) how long it takes you to pass each mark as you walk at a pace you try to keep constant.

Phase 2. Data processing and graphs: Construct a table with your position ($s$) and time ($t$) data. Plot the position-time ($s-t$) graph and calculate the average speed for each section to assess whether the slope remains constant.

Phase 3. Final report: Write a brief one-page report where you explain the difference between your trajectory and your displacement with justification, and conclude whether you achieved true Uniform Rectilinear Motion, analysing the influence of measurement errors.

Evaluation criteria:

• Correct definition and application of position, trajectory, and displacement.

• Rigorous data recording with uncertainties and precise creation of the table and $s-t$ graph.

• Mathematical and conceptual justification of whether the motion corresponds to URM by calculating speed.