Exercises: Uniform linear motion

An airport moving walkway moves in a straight line and always advances at the same speed.

1. a) What type of motion does it describe?
2. b) State the two conditions that motion must satisfy to be uniform and linear.

A pedestrian moves with uniform linear motion according to the equation $x = x_0 + v \times t$, with $x_0 = 10\ \text{m}$ and $v = 2\ \text{m/s}$. Calculate their position when $t = 5\ \text{s}$.

The position of a cyclist is given, in International System units, by the equation $x = 5 + 3 \times t$.

1. a) What is their initial position $x_0$ and their velocity $v$?
2. b) Calculate their position when $t = 10\ \text{s}$.

The position of a scooter, in International System units, is $x = 50 - 4 \times t$.

1. a) Explain what the minus sign on the velocity means: is it moving away from or towards the origin?
2. b) Calculate its position when $t = 5\ \text{s}$.

A cyclist leaves the origin and pedals in a straight line at a constant speed of $6\ \text{m/s}$.

1. a) What is their position after $20\ \text{s}$?
2. b) What distance have they covered in that time? Explain why it coincides with their position.

A luggage conveyor belt moves at $0.5\ \text{m/s}$. If the belt is $30\ \text{m}$ long, how long does it take a suitcase to travel from end to end?

Two runners describe their motion, in International System units, using the equations $x = 8 + 2 \times t$ and $x = 8 + 5 \times t$. The $x$-$t$ graphs for both are shown in the figure.

1. a) What do the two motions have in common, and how do they differ?
2. b) Calculate the position of each runner when $t = 4\ \text{s}$.

A car travels on a motorway at a constant speed of $90\ \text{km/h}$. What distance does it cover in $3\ \text{h}$?

A runner maintains a constant speed of $18\ \text{km/h}$.

1. a) Express their speed in $\text{m/s}$.
2. b) How long does it take them to cover $200\ \text{m}$?

A train passes through a station without stopping, at $108\ \text{km/h}$, and maintains that speed. We take the station as the origin.

1. a) Express the speed in $\text{m/s}$.
2. b) Write the equation for its position.
3. c) Calculate its position $45\ \text{s}$ after passing through the station.

The position of a cyclist, in International System units, is $x = 20 + 5 \times t$.

1. a) What is their position when $t = 6\ \text{s}$?
2. b) At what instant do they pass through position $x = 120\ \text{m}$?

A toy car moves according to the equation $x = 200 - 10 \times t$, in International System units.

1. a) At what instant does it reach the origin?
2. b) What is its position when $t = 12\ \text{s}$?

You leave home walking at $1.5\ \text{m/s}$ towards the bus stop, which is $450\ \text{m}$ away.

1. a) How many seconds does it take you to arrive?
2. b) Express the result in minutes.

You shout in front of a vertical wall that is $170\ \text{m}$ away and hear the echo. Sound travels through air at $340\ \text{m/s}$. The situation is shown in the figure.

1. a) What distance does the sound travel from the moment you shout until you hear the echo? Remember that it travels there and back.
2. b) How long does it take for the echo to reach you?

Along a straight cycle path, two people are travelling in the same direction: A at $x_A = 5 \times t$ and B at $x_B = 60 + 2 \times t$, in International System units. The initial situation is shown in the figure.

1. a) Calculate the position of each person when $t = 8\ \text{s}$.
2. b) What distance separates them at that instant?

Light travels in a vacuum at $3 \times 10^{8}\ \text{m/s}$ and the average distance from the Sun to the Earth is $1.5 \times 10^{11}\ \text{m}$. The diagram is shown in the figure.

1. a) Calculate the time it takes for sunlight to reach us, in seconds.
2. b) Express it in minutes, to one decimal place.

Two cities are connected by a straight railroad line of $300\ \text{km}$. A train leaves city A for B at $90\ \text{km/h}$ and, at the same time, another leaves city B for A at $60\ \text{km/h}$. The situation is shown in the figure.

1. a) How many kilometers do they get closer to each other every hour?
2. b) How long do they take to cross paths?
3. c) At what distance from city A do they cross paths?

A cyclist leaves a point at $6\ \text{m/s}$. Ten seconds later, a scooter leaves from the same point in the same direction at $10\ \text{m/s}$.

1. a) Write the position equation for each vehicle taking $t = 0$ when the scooter starts. Remember that when the scooter starts, the cyclist already has a $10\ \text{s}$ head start.
2. b) How long does it take for the scooter to catch up with the cyclist?
3. c) At what distance from the starting point does it catch up?

Ana leaves her house walking at $4\ \text{m/s}$. Twenty seconds later, Bruno leaves the same house in the same direction, running at $6\ \text{m/s}$. The situation at the instant Bruno leaves is shown in the figure.

1. a) Write the position equations for both, counting time from when Bruno leaves.
2. b) Calculate the time it takes for Bruno to catch up with Ana.
3. c) At what distance from house does he catch up?
4. d) Check who is ahead $10\ \text{s}$ after Bruno leaves and explain the result.

Two objects move along the same straight line with equations, in International System units, $x_A = 5 \times t$ and $x_B = 90 - 4 \times t$.

1. a) What distance separates them initially? Are they getting closer or further apart? Justify your answer using the signs of the velocities.
2. b) Calculate the instant at which they cross paths.
3. c) At what position do they do so?

A truck leaves a town along a straight divided highway at $72\ \text{km/h}$. Half an hour later, a car leaves the same point at $108\ \text{km/h}$, heading in the same direction.

1. a) Express both speeds in $\text{m/s}$.
2. b) Write the position equations for both vehicles, counting time from when the car leaves.
3. c) Calculate the time it takes for the car to catch up with the truck, expressed in hours.
4. d) At what distance from the town does it catch up, in kilometers?

On a test track, a motorbike leaves the origin at $25\ \text{m/s}$ at the same time as a car located $500\ \text{m}$ ahead starts off and travels at $15\ \text{m/s}$ in the same direction. The initial situation is shown in the figure.

1. a) Write the position equations for both vehicles.
2. b) Calculate the instant at which the motorbike catches up with the car.
3. c) At what position does it catch up? Check your result by substituting the time into both equations.

Two athletes train on a straight track with equations, in International System units, $x_A = 2 + 4 \times t$ and $x_B = 10 + 2 \times t$.

1. a) Which of the two starts further ahead, and with how many meters head start?
2. b) Calculate the instant and position where one catches up with the other.
3. c) Determine the position of each athlete when $t = 10\ \text{s}$ and the distance between them.
4. d) Explain, based on their speeds, why the one who started behind ends up taking the lead.

Two objects describe their motion, in International System units, using the equations $x_A = 3 + 5 \times t$ and $x_B = 10 + 5 \times t$. The $x$-$t$ graphs for both are shown in the figure.

1. a) Try to calculate the instant at which they would meet by equating the two equations, and explain what happens.
2. b) What distance separates them at any instant? Prove it by calculating their positions at two different instants.
3. c) State the condition that the speeds of two objects undergoing uniform linear motion must satisfy in order for them to meet.