Exercises: Interpretation of x-t and v-t graphs

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 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.

Calculate their speed.

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 $10\ \text{s}$?

Match each description of an x-t graph with the corresponding motion and justify your choices:

1. a) A horizontal straight line.
2. b) A sloped straight line rising upwards.
3. c) A curve getting progressively steeper.

Motions: it is stationary; it moves at a constant speed; it moves faster and faster.

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 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.

What information does the gradient of a position-time graph give? And that of a velocity-time graph? Answer in your own words and provide a numerical example for each.

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 $(60\ \text{min},\ 16\ \text{km})$.

1. a) Calculate the speed of each section, in $\text{km/min}$.
2. b) What happens in the middle section? What do you think the cyclists did during that time?

The v-t graph of a car has three sections, as shown in the figure: from $(0\ \text{s},\ 0\ \text{m/s})$ to $(10\ \text{s},\ 20\ \text{m/s})$; horizontal at $20\ \text{m/s}$ up to $t = 30\ \text{s}$; and from $(30\ \text{s},\ 20\ \text{m/s})$ to $(35\ \text{s},\ 0\ \text{m/s})$.

1. a) Calculate the acceleration of each section.
2. b) Describe in words what the car does throughout the entire journey.

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.

1. a) Calculate each person's speed.
2. b) Who started ahead? Who runs faster?
3. 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 opposite direction, passes through $(0\ \text{s},\ 800\ \text{m})$ and $(40\ \text{s},\ 0\ \text{m})$, as shown in the figure.

1. a) Calculate the velocity of each train, including its sign.
2. b) The two lines intersect at a point: explain what this point means.
3. c) Calculate the time and position at which the trains pass each other.

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.

1. a) What distance does it travel in the first section? And in the second?
2. b) What total distance does it travel in $40\ \text{s}$?
3. c) What is its average speed over the entire journey?

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.

1. a) Does its speed increase, decrease or stay constant? Reason your answer based on how the gradient changes.
2. b) What would the curve look like if, instead, the car braked to a stop? Describe it in words.

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.

1. a) Calculate their acceleration.
2. b) What does the sign of this acceleration mean?
3. c) What would it mean if the line continued below the axis, with negative velocities?

A car waits stationary at a traffic light; when it turns green, it pulls away gaining speed uniformly, travels for a while at constant speed and finally brakes to a stop. Which of these descriptions of v-t graphs corresponds to the car? Justify your choice and explain why the others are ruled out:

1. a) A horizontal line above zero at all times.
2. b) Starting from zero: a line that rises, then a horizontal section, and then a line that falls to zero.
3. c) A line that falls from a high value to zero and stays at zero.

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 $(55\ \text{min},\ 0\ \text{km})$.

1. a) Calculate the speed of each section, in $\text{km/min}$.
2. b) What does the last section mean, whose gradient is negative?
3. c) In which section is the bus travelling fastest? Justify it by comparing the gradients.

The v-t graph of a car has these sections, as shown in the figure: it rises in a straight line from $(0\ \text{s},\ 0\ \text{m/s})$ to $(5\ \text{s},\ 10\ \text{m/s})$; remains at $10\ \text{m/s}$ up to $t = 25\ \text{s}$; and falls in a straight line to $(35\ \text{s},\ 0\ \text{m/s})$.

1. a) Calculate the acceleration of each section.
2. b) Calculate the distance travelled in the constant speed section.
3. c) In the sections with acceleration, the distance corresponds to the area of the triangle under the graph. Calculate the distance of the first and third sections using this idea and find the total distance of the journey.

On an x-t graph, the line of a cyclist passes through $(0\ \text{s},\ 0\ \text{m})$ and $(30\ \text{s},\ 300\ \text{m})$, and that of a pedestrian, travelling in the opposite direction, passes through $(0\ \text{s},\ 500\ \text{m})$ and $(50\ \text{s},\ 0\ \text{m})$, as shown in the figure.

1. a) Calculate the velocity of each, including their sign.
2. b) Write the position equation for each.
3. c) The point where the lines intersect is the meeting point: calculate when and where it occurs.
4. d) Check the result by substituting into both equations.

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 $(5\ \text{s},\ 20\ \text{m})$.

1. a) Calculate the speed of the first section.
2. b) Explain why the second section is impossible in reality: notice how time progresses.
3. 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.

Two cars leave a traffic light at the same time. The v-t graph for car A rises in a straight line from $(0\ \text{s},\ 0\ \text{m/s})$ to $(10\ \text{s},\ 20\ \text{m/s})$. That of car B rises from $(0\ \text{s},\ 0\ \text{m/s})$ to $(5\ \text{s},\ 20\ \text{m/s})$ and then remains at $20\ \text{m/s}$ up to $t = 10\ \text{s}$, as shown in the figure.

1. a) Calculate the acceleration of each car as it starts moving.
2. b) By calculating the area under each graph, find the distance travelled by each car in those $10\ \text{s}$.
3. c) Which one is ahead upon reaching $10\ \text{s}$? Justify this using your results.

The v-t graph of a rising lift has these sections, as shown in the figure: from $(0\ \text{s},\ 0\ \text{m/s})$ to $(2\ \text{s},\ 4\ \text{m/s})$; horizontal at $4\ \text{m/s}$ up to $t = 8\ \text{s}$; and from $(8\ \text{s},\ 4\ \text{m/s})$ to $(10\ \text{s},\ 0\ \text{m/s})$.

1. a) Calculate the acceleration of each section.
2. b) Calculate the distance for each section from the area under the graph, and the total height the lift ascends.
3. c) Describe in words what the x-t graph of this lift would look like: in which sections would it be a curve and in which a straight line? How does its gradient change? Justify your answer.

On an x-t graph, a cyclist appears as a straight line from $(0\ \text{s},\ 0\ \text{m})$ to $(60\ \text{s},\ 300\ \text{m})$. A motorbike starts later and its line passes through $(20\ \text{s},\ 0\ \text{m})$ and $(40\ \text{s},\ 300\ \text{m})$, as shown in the figure.

1. a) Calculate the speed of each.
2. b) Write their position equations (remember that the motorbike starts at $t = 20\ \text{s}$).
3. c) At what time and position does the motorbike catch up with the cyclist?
4. d) Who is ahead at $t = 25\ \text{s}$? Justify your answer with calculations.

The v-t graph of a $100\ \text{m}$ final shows two athletes, as shown in the figure: Vera rises in a straight line from $(0\ \text{s},\ 0\ \text{m/s})$ to $(4\ \text{s},\ 10\ \text{m/s})$ and then maintains $10\ \text{m/s}$ until the end; Hugo rises from $(0\ \text{s},\ 0\ \text{m/s})$ to $(2\ \text{s},\ 10\ \text{m/s})$ and also maintains $10\ \text{m/s}$.

1. a) Calculate the initial acceleration of each athlete.
2. b) Calculate the distance travelled by each at $10\ \text{s}$ (areas under the graph).
3. c) If both maintain that maximum speed, who crosses the $100\ \text{m}$ finish line first and at what time? Show reasoning for all calculations.

Topic challenge. Become a road safety expert by analysing the motion of a vehicle!

1. Data collection: Build a gentle ramp using cardboard or a board on the floor, with a length of $1,0\text{ m}$. Mark distances with tape at $0,25\text{ m}$, $0,50\text{ m}$, $0,75\text{ m}$, and $1,00\text{ m}$. Release a ball or toy car from rest and time how long it takes to reach each mark using your phone ($t \pm 0,2\text{ s}$). Repeat three times for each mark and record the mean times in a table.

2. Kinematic analysis: Add to your table the calculation of the velocity in each section and the acceleration ($a = 2x/t^2$). Based on the constancy of these parameters, deduce and justify whether the descent best fits uniform rectilinear motion (MRU) or uniformly accelerated rectilinear motion (MRUA).

3. Final product: Prepare a brief report containing the data table with its uncertainties, the $x-t$ and $v-t$ graphs drawn to scale, and a text interpreting the shape of the lines and the physical meaning of their slopes.

Assessment criteria:

• Experimental rigour in data collection, calculation of means, and expression of uncertainties.

• Correct plotting and interpretation of the $x-t$ and $v-t$ graphs.

• Coherent kinematic justification for classifying the motion as MRU or MRUA based on the obtained velocity and acceleration.