Constant speed in a straight line
An Amtrak train travels along a straight stretch of track between two stations $12\ \text{km}$ apart. The driver holds a steady $72\ \text{km/h}$ the whole way. Could you work out, without being on board, how long the journey takes? Motion in a straight line at constant speed is easy to predict. We will come back to the train at the end.
What constant speed in a straight line means
If you watch a bubble rising up a vertical tube of oil and note down its height every second ($0\ \text{s} \rightarrow 0\ \text{cm}$, $1\ \text{s} \rightarrow 4\ \text{cm}$, $2\ \text{s} \rightarrow 8\ \text{cm}$, $3\ \text{s} \rightarrow 12\ \text{cm}$), you will see that it covers identical distances in equal times: how do we define and describe this very regular motion mathematically?
Explain in your own words what uniform motion in a straight line is: what is its trajectory like? What is its speed like?
Explain in your own words what uniform motion in a straight line is: what is its trajectory like? What is its speed like?
Which of these motions can be uniform rectilinear motion (constant velocity in a straight line)? Justify your answers. a) A train traveling along a straight track at…
Which of these motions can be uniform rectilinear motion (constant velocity in a straight line)? Justify your answers.
- a) A train traveling along a straight track at $120\ \text{km/h}$ without changing its speed.
- b) A car stuck in a traffic jam.
- c) A luggage conveyor belt that turns on curves.
- d) An elevator ascending at a constant speed.
Complete the sentence with the missing words: "In uniform motion in a straight line, the trajectory is a $\dots$ line and the speed is $\dots$". Then give a real-life…
Complete the sentence with the missing words: "In uniform motion in a straight line, the trajectory is a $\dots$ line and the speed is $\dots$". Then give a real-life example.
The conveyor belt at a supermarket checkout moves the groceries $2\ \text{m}$ in $10\ \text{s}$. At what speed is the belt moving?
The conveyor belt at a supermarket checkout moves the groceries $2\ \text{m}$ in $10\ \text{s}$. At what speed is the belt moving?
An airplane flies in a straight line and covers $150\ \text{km}$ in $0.5\ \text{h}$ at constant speed. What is its speed in $\text{km/h}$?
An airplane flies in a straight line and covers $150\ \text{km}$ in $0.5\ \text{h}$ at constant speed. What is its speed in $\text{km/h}$?
Distance from the start
In this graph, the sliders let you change the starting distance and the speed. Move the slider for the starting distance (the distance from the start point at time 0, in m) and the slider for the speed (in m/s) and see how the line on the distance–time graph (distance from start in m against time in s) changes: the slope of the line is the speed.
An electric scooter travels $60\ \text{m}$ in $12\ \text{s}$ along a straight street, without changing its speed. What is its speed?
An electric scooter travels $60\ \text{m}$ in $12\ \text{s}$ along a straight street, without changing its speed. What is its speed?
An object moving at constant velocity (uniform rectilinear motion) covers $5\ \text{m}$ every second. a) How many meters does it cover in $4\ \text{s}$? b) And in…
An object moving at constant velocity (uniform rectilinear motion) covers $5\ \text{m}$ every second.
- a) How many meters does it cover in $4\ \text{s}$?
- b) And in $10\ \text{s}$?
A tortoise moves in a straight line at $0.3\ \text{m/s}$ for $10\ \text{s}$. What distance does it cover?
A tortoise moves in a straight line at $0.3\ \text{m/s}$ for $10\ \text{s}$. What distance does it cover?
How long does it take a car to cover $240\ \text{km}$ on a highway if it maintains a constant speed of $80\ \text{km/h}$?
How long does it take a car to cover $240\ \text{km}$ on a highway if it maintains a constant speed of $80\ \text{km/h}$?
A runner maintains a constant speed of $5\ \text{m/s}$. How long does it take them to cover $900\ \text{m}$? Give the answer in seconds and in minutes.
A runner maintains a constant speed of $5\ \text{m/s}$. How long does it take them to cover $900\ \text{m}$? Give the answer in seconds and in minutes.
A construction hoist rises at $0.5\ \text{m/s}$. How long does it take to get from the ground to scaffolding $12\ \text{m}$ high?
A construction hoist rises at $0.5\ \text{m/s}$. How long does it take to get from the ground to scaffolding $12\ \text{m}$ high?
An object passes through the positions $x = 0\ \text{m}$, $4\ \text{m}$, $8\ \text{m}$ and $12\ \text{m}$ at times $t = 0\ \text{s}$, $1\ \text{s}$, $2\ \text{s}$ and…
An object passes through the positions $x = 0\ \text{m}$, $4\ \text{m}$, $8\ \text{m}$ and $12\ \text{m}$ at times $t = 0\ \text{s}$, $1\ \text{s}$, $2\ \text{s}$ and $3\ \text{s}$, as shown in the figure.
- a) What distance does it cover each second?
- b) What is its speed? Is it uniform rectilinear motion (constant velocity)?
- c) If it continues like this, what position will it be at time $t = 10\ \text{s}$?
Units and time
In science we use $\text{m/s}$ or $\text{km/h}$, while road signs in the U.S. show $\text{mph}$. Times in seconds, minutes or hours. Before you calculate, make sure the units match. The same method works with miles and mph: a bus that travels $25\ \text{mi}$ at a constant $30\ \text{mph}$ takes $25 \div 30 = \frac{5}{6}\ \text{h}$, which is $50\ \text{min}$.
A cyclist goes down a mountain pass at $72\ \text{km/h}$. a) Express their speed in $\text{m/s}$. b) What distance do they cover in $15\ \text{s}$ at that speed?
A cyclist goes down a mountain pass at $72\ \text{km/h}$.
- a) Express their speed in $\text{m/s}$.
- b) What distance do they cover in $15\ \text{s}$ at that speed?
Between two towns there are $30\ \text{km}$ of straight road. a) How long does a car traveling at $60\ \text{km/h}$ take? Give the answer in hours and in minutes. b) At…
Between two towns there are $30\ \text{km}$ of straight road.
- a) How long does a car traveling at $60\ \text{km/h}$ take? Give the answer in hours and in minutes.
- b) At what speed would it have to travel to take only $20\ \text{min}$?
A boat sails in a straight line at $18\ \text{km/h}$ for $30\ \text{min}$. What distance does it cover?
A boat sails in a straight line at $18\ \text{km/h}$ for $30\ \text{min}$. What distance does it cover?
Sunlight travels $1.5\times10^{8}\ \text{km}$ to Earth at a speed of $3\times10^{5}\ \text{km/s}$. a) How many seconds does it take for sunlight to reach us? b) Express…
Sunlight travels $1.5\times10^{8}\ \text{km}$ to Earth at a speed of $3\times10^{5}\ \text{km/s}$.
- a) How many seconds does it take for sunlight to reach us?
- b) Express the answer in minutes and seconds.
- c) If the Sun were to suddenly go out, when would we see it from Earth? Explain your reasoning.
A bus departs at 10:00 and needs to travel $45\ \text{km}$ at a constant speed of $60\ \text{km/h}$. a) What time does it arrive at its destination? b) If a traffic jam…
A bus departs at 10:00 and needs to travel $45\ \text{km}$ at a constant speed of $60\ \text{km/h}$.
- a) What time does it arrive at its destination?
- b) If a traffic jam forces it to travel at $45\ \text{km/h}$, what time would it arrive?
- c) How many minutes late is it in that case?
Average speed
Real journeys are rarely at constant speed. If the distances in equal times are different, the speed is not constant, but we can still describe the whole trip by its average speed.
Measuring every second, a car travels $10\ \text{m}$ in the first second, $15\ \text{m}$ in the second second and $20\ \text{m}$ in the third second. a) Is it uniform…
Measuring every second, a car travels $10\ \text{m}$ in the first second, $15\ \text{m}$ in the second second and $20\ \text{m}$ in the third second.
- a) Is it uniform motion in a straight line? Why?
- b) What is its average speed during those $3\ \text{s}$?
A family can go to the beach via two routes: by highway, $120\ \text{km}$ at $100\ \text{km/h}$, or by main road, $90\ \text{km}$ at $60\ \text{km/h}$. a) Calculate the…
A family can go to the beach via two routes: by highway, $120\ \text{km}$ at $100\ \text{km/h}$, or by main road, $90\ \text{km}$ at $60\ \text{km/h}$.
- a) Calculate the time they take for each route, in hours.
- b) Express those times in minutes.
- c) Which route gets them there faster? How many minutes difference is there?
Meeting and chasing
Moving surfaces
Speed depends on what you measure it against. On a moving walkway, if you walk the same way as the belt, add the speeds; if you walk the opposite way, subtract them. The sign of the result tells you the direction you move relative to the floor.
At the gym, the treadmill displays $12\ \text{km/h}$. a) How many meters does a runner "cover" in $5\ \text{min}$ on the treadmill? b) Has she changed position relative…
At the gym, the treadmill displays $12\ \text{km/h}$.
- a) How many meters does a runner "cover" in $5\ \text{min}$ on the treadmill?
- b) Has she changed position relative to the gym floor? Explain the difference between motion relative to the treadmill and relative to the floor.
Back to the start: the train covers $12\ \text{km}$ at $72\ \text{km/h}$, so it takes $12 \div 72 = \frac{1}{6}\ \text{h}$, which is $10\ \text{min}$.