Uniform Rectilinear Motion
Santo Domingo and Santiago are about 155 km apart along the Duarte Highway, an almost straight road. If a car always keeps the same speed, how can you tell how long it will take and where it will be at each moment? That very regular motion has its own name and a very simple equation.
What URM Is
Watch a bubble rising through a tube of oil. You write down its height every second: $0\ \text{s} \rightarrow 0\ \text{cm}$, $1\ \text{s} \rightarrow 4\ \text{cm}$, $2\ \text{s} \rightarrow 8\ \text{cm}$, $3\ \text{s} \rightarrow 12\ \text{cm}$. It covers equal distances in equal times. What is this motion called, and how is it described?
Explain in your own words what uniform motion in a straight line (MRU) is: what is its trajectory like? What is its speed like?
Explain in your own words what uniform motion in a straight line (MRU) is: what is its trajectory like? What is its speed like?
Which of these motions can be a MRU? Justify your answers. a) A train travelling along a straight track at $120\ \text{km/h}$ without changing its speed. b) A car stuck…
Which of these motions can be a MRU? Justify your answers.
- a) A train travelling 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) A lift ascending at a constant 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 $3\ \text{s}$?
Calculating Speed, Distance, and Time
If the speed doesn't change, one division is enough to find it: distance divided by time. For example, a skateboard that covers $60\ \text{m}$ in $12\ \text{s}$ goes at $60 / 12 = 5\ \text{m/s}$. The other two quantities come from that same relationship.
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 aeroplane 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 aeroplane flies in a straight line and covers $150\ \text{km}$ in $0.5\ \text{h}$ at constant speed. What is its speed in $\text{km/h}$?
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 villages there are $30\ \text{km}$ of straight road. a) How long does a car travelling at $60\ \text{km/h}$ take? Give the answer in hours and in minutes. b)…
Between two villages there are $30\ \text{km}$ of straight road.
- a) How long does a car travelling 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 $20\ \text{min}$?
The URM Equation
To know where the object is at each moment, you have to take into account where it started.
Move the sliders for the initial position $x_0$ and the speed $v$ and watch how the line on the graph $x(t)$ changes.
An object moving with MRU covers $5\ \text{m}$ every second. a) How many meters does it cover in $4\ \text{s}$? b) And in $10\ \text{s}$?
An object moving with MRU 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?
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 a MRU?
- c) If it continues like this, what position will it be at time $t = 10\ \text{s}$?
Meetings and Chases
Two cyclists leave at the same time, one from each village, along the straight $24\ \text{km}$ road connecting them. One travels at $15\ \text{km/h}$ and the other at…
Two cyclists leave at the same time, one from each village, along the straight $24\ \text{km}$ road connecting them. One travels at $15\ \text{km/h}$ and the other at $9\ \text{km/h}$, as shown in the figure.
- a) At what speed do they approach each other?
- b) How long does it take for them to meet?
- c) What distance has each travelled when they meet?
Leo sets off walking along a straight path at $1.5\ \text{m/s}$. Sixty seconds later Nora sets off along the same path at $3\ \text{m/s}$, as shown in the figure (at the…
Leo sets off walking along a straight path at $1.5\ \text{m/s}$. Sixty seconds later Nora sets off along the same path at $3\ \text{m/s}$, as shown in the figure (at the moment Nora sets off).
- a) How much of a head start does Leo have at the moment Nora sets off?
- b) How many meters does Nora gain on Leo each second?
- c) How long does it take Nora to catch up with him?
Problems with More Than One Motion
Here you apply everything above: comparing routes, calculating arrival times, average speed, and motions that add together.
How long does it take a car to cover $240\ \text{km}$ on a motorway 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 motorway 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 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?
An escalator ascends at $0.5\ \text{m/s}$, as shown in the figure. a) If you walk up it at $1\ \text{m/s}$ relative to the steps, at what speed are you ascending…
An escalator ascends at $0.5\ \text{m/s}$, as shown in the figure.
- a) If you walk up it at $1\ \text{m/s}$ relative to the steps, at what speed are you ascending relative to the shopping center floor?
- b) What if you walk down at $1\ \text{m/s}$ while the escalator is ascending? In which direction are you moving relative to the floor?
- c) What would happen if you walked down at $0.5\ \text{m/s}$? Explain each case.
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? Reason your answer.
A family can go to the beach via two routes: by motorway, $120\ \text{km}$ 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 motorway, $120\ \text{km}$ 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?
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?
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.
A fishing boat sails in a straight line at $10\ \text{km/h}$ for $3\ \text{h}$. a) How far from the port is it? b) At what speed must it return to be back in…
A fishing boat sails in a straight line at $10\ \text{km/h}$ for $3\ \text{h}$.
- a) How far from the port is it?
- b) At what speed must it return to be back in $2\ \text{h}$?
- c) What is the average speed for the whole trip, there and back?
Back to the beginning: if a car keeps $80\ \text{km/h}$ on the Duarte Highway, it takes $155/80 \approx 1.9\ \text{h}$ to go from Santo Domingo to Santiago, and its position at each moment comes from $x = x_0 + v \times t$.