Exercises: Uniform rectilinear motion

Explain what is meant by uniform rectilinear motion (motion in a straight line at constant velocity) and describe two real-life examples of motion that, at least for a while, can be considered uniform rectilinear motion.

A walker starts from position $x_0 = 10\ \text{m}$ and moves forward in a straight line at $2\ \text{m/s}$, as shown in the figure.

Using the equation $x = x_0 + v\times t$, calculate her position after $5\ \text{s}$.

A runner starts from the origin ($x_0 = 0\ \text{m}$) and runs at a constant speed of $4\ \text{m/s}$. What position are they at after $10\ \text{s}$?

A toy cart moving at constant velocity (uniform rectilinear motion) passes position $5\ \text{m}$ at time $t = 0\ \text{s}$ and position $45\ \text{m}$ at $t = 10\ \text{s}$. What is its velocity?

A robot vacuum cleaner moves forward in a straight line at $0.5\ \text{m/s}$. How long does it take to cover the $6\ \text{m}$ hallway?

A cyclist starts from kilometer marker $12$ on a greenway and pedals to kilometer marker $32$, as shown in the figure.

What is their final position? What distance have they covered? Explain the difference between the two concepts.

The position of a train is given by the equation $x = 20 + 5\times t$, in SI units. From what position does it start? At what velocity is it moving? Where will it be at time $t = 10\ \text{s}$?

An escalator moves up at a constant speed of $0.6\ \text{m/s}$. At what position is a person $15\ \text{s}$ after passing the origin?

A bus leaves kilometer marker $50$ on a straight highway and travels at $80\ \text{km/h}$.

1. a) At which kilometer marker is it located after $1.5\ \text{h}$?
2. b) What distance has it covered?

A hiker starts from position $x_0 = 20\ \text{m}$ and walks at $4\ \text{m/s}$. How long does it take them to reach position $x = 100\ \text{m}$?

A cyclist travels at $72\ \text{km/h}$ along a straight road. Express their speed in $\text{m/s}$ and calculate what position they will be at after $30\ \text{s}$ if they started from the origin.

A cyclist leaves the origin at $8\ \text{m/s}$. At the same time, his friend starts $30\ \text{m}$ ahead, in the same direction, at $5\ \text{m/s}$, as shown in the figure.

1. a) Write down the position equation for each person.
2. b) How long does it take the cyclist to catch up with his friend?

Two ships set sail at the same time from two ports $120\ \text{km}$ apart, sailing toward each other at $25\ \text{km/h}$ and $15\ \text{km/h}$.

1. a) How many kilometers closer to each other do they get every hour?
2. b) How long does it take for them to meet?
3. c) At what distance from the first port do they meet?

A drone is at position $x_0 = 100\ \text{m}$ and flies toward the origin at $4\ \text{m/s}$, so that its velocity is negative: $v = -4\ \text{m/s}$, as shown in the figure.

1. a) Write down its position equation.
2. b) What is its position at time $t = 10\ \text{s}$?
3. c) At what time does it pass through the origin?

A train undergoing constant-velocity motion (uniform rectilinear motion) is at position $14\ \text{m}$ when $t = 2\ \text{s}$ and at position $30\ \text{m}$ when $t = 6\ \text{s}$.

1. a) Calculate its velocity.
2. b) What was its initial position, at $t = 0\ \text{s}$?
3. c) Write its position equation.

A ship travels along a straight canal at $18\ \text{km/h}$. Express its speed in $\text{m/s}$ and calculate how many meters it advances in $5\ \text{min}$.

Two trains depart at the same time from two stations separated by $240\ \text{km}$, one from each station and traveling toward each other, at $100\ \text{km/h}$ and $60\ \text{km/h}$, as shown in the figure.

1. a) Write down the position equation for each train, taking the first station as the origin.
2. b) Calculate at what time and at what position they cross paths.
3. c) Check the result by substituting the time into both equations.

In a charity race, Emma runs at $4\ \text{m/s}$ and has a $60\ \text{m}$ head start over Liam, who runs at $6\ \text{m/s}$. The finish line is $200\ \text{m}$ ahead of Emma, as shown in the figure.

1. a) Write down the position equation for each runner, taking Liam's position as the origin.
2. b) Calculate at what time and at what position Liam catches up with Emma.
3. c) Does he manage to catch up with her before the finish line? Justify your answer using calculations.

A ship leaves port at $10\ \text{km/h}$. Two hours later, another ship sets off in the same direction at $15\ \text{km/h}$.

1. a) What distance has the first ship covered when the second ship sets off?
2. b) Write the position equations for both ships (careful: the second one leaves $2\ \text{h}$ later).
3. c) How long does it take for the second ship to catch up with the first ship, counting from its own departure time? At what distance from the port?

A car travels along a straight line and its position follows the equation $x = 40 + 12\times t$, in SI units.

1. a) Where was it at $t = 0\ \text{s}$ and what is its velocity?
2. b) What position will it be in after $20\ \text{s}$?
3. c) At what time does it pass through position $280\ \text{m}$?
4. d) If another car follows the equation $x = 300 - 8\times t$, at what time and at what position do the two cross paths?

On an airport walkway moving forward at $0.5\ \text{m/s}$, a passenger walks at $1.5\ \text{m/s}$ relative to the walkway, in the same direction, as shown in the figure.

1. a) At what speed does the passenger move relative to the airport floor?
2. b) The walkway is $60\ \text{m}$ long: how long does it take to walk along it?
3. c) Another passenger walks on the floor alongside the walkway at $1.5\ \text{m/s}$. Who takes less time to cover the $60\ \text{m}$? What is the difference in seconds?

The position of a cyclist is given by the equation $x = 50 + 10\times t$, in SI units.

1. a) What position does the cyclist occupy at $t = 2\ \text{s}$? And at $t = 4\ \text{s}$?
2. b) What distance does the cyclist cover between those two moments?
3. c) Your classmate says: "at $t = 4\ \text{s}$ twice as much time has passed as at $t = 2\ \text{s}$, so the cyclist is twice as far from the origin." Is she right? Explain her error using the previous parts.

In a park, there is a straight path of $300\ \text{m}$ between two fountains. Delia leaves the first fountain at $3\ \text{m/s}$ and, at the same time, Eloy leaves the second at $2\ \text{m/s}$, each heading toward the other's fountain.

1. a) Write the position equation for each person, taking the first fountain as the origin.
2. b) At what time and at what position do they cross paths?
3. c) How long after crossing paths does Delia reach the second fountain?

A car travels along a highway at $108\ \text{km/h}$ and a truck is $500\ \text{m}$ ahead, in the same direction, traveling at $72\ \text{km/h}$.

1. a) Express both speeds in $\text{m/s}$.
2. b) Write down the position equations taking the initial position of the car as the origin.
3. c) Calculate how long it takes for the car to catch up with the truck and what distance it has covered by then.