Acceleration
A bus pulls out of the Metrobús stop: it doesn't keep the same speed, but gets a little faster every second. We call this quantity, which measures how fast velocity changes, acceleration. Going fast is not the same as accelerating a lot, and telling them apart is the key to this section.
What acceleration is
You look at the speedometer of a motorcycle starting off in a straight line: at $0\ \text{s}$ it reads $0\ \text{m/s}$, at $1\ \text{s}$ it reads $4\ \text{m/s}$, at $2\ \text{s}$ it reads $8\ \text{m/s}$, and at $3\ \text{s}$ it reads $12\ \text{m/s}$. How do we describe the rate at which that velocity changes with a single quantity?
Every second the velocity grows by the same amount: $4\ \text{m/s}$. That change per second is the acceleration.
If you know the acceleration and the time, you can calculate the final velocity: since it gains $a$ each second, $v = v_0 + a \cdot t$. With $v_0 = 0$ it becomes $v = a \cdot t$.
A car has an acceleration of $3\ \text{m/s}^2$. Explain what this value means: by how much does its speed increase each second? If it starts from rest, what speed will…
A car has an acceleration of $3\ \text{m/s}^2$. Explain what this value means: by how much does its speed increase each second? If it starts from rest, what speed will it have after $4\ \text{s}$?
A sports car goes from $0$ to $90\ \text{km/h}$ in $5\ \text{s}$. Express the final speed in $\text{m/s}$ and calculate the acceleration in $\text{m/s}^2$.
A sports car goes from $0$ to $90\ \text{km/h}$ in $5\ \text{s}$. Express the final speed in $\text{m/s}$ and calculate the acceleration in $\text{m/s}^2$.
A rocket-powered sled starts from rest with an acceleration of $4\ \text{m/s}^2$ for $10\ \text{s}$. a) What speed does it reach, in $\text{m/s}$? b) Express it in…
A rocket-powered sled starts from rest with an acceleration of $4\ \text{m/s}^2$ for $10\ \text{s}$.
- a) What speed does it reach, in $\text{m/s}$?
- b) Express it in $\text{km/h}$.
- c) If it then brakes with an acceleration of $-2\ \text{m/s}^2$, how long does it take to come to a stop?
Acceleration and braking: interpreting the signs
When the bus driver brakes on arriving at the station, the bus's velocity decreases every second. What acceleration does it have then?
A car brakes and goes from $20\ \text{m/s}$ to a standstill in $4\ \text{s}$. Calculate its acceleration. Why is the result negative?
A car brakes and goes from $20\ \text{m/s}$ to a standstill in $4\ \text{s}$. Calculate its acceleration. Why is the result negative?
A tram travelling at $72\ \text{km/h}$ brakes to a standstill in $8\ \text{s}$. Express the initial speed in $\text{m/s}$ and calculate the braking acceleration.
A tram travelling at $72\ \text{km/h}$ brakes to a standstill in $8\ \text{s}$. Express the initial speed in $\text{m/s}$ and calculate the braking acceleration.
A racing car goes from $0$ to $30\ \text{m/s}$ in $5\ \text{s}$ and, on reaching the bend, brakes from $30\ \text{m/s}$ to $10\ \text{m/s}$ in $4\ \text{s}$, as shown in…
A racing car goes from $0$ to $30\ \text{m/s}$ in $5\ \text{s}$ and, on reaching the bend, brakes from $30\ \text{m/s}$ to $10\ \text{m/s}$ in $4\ \text{s}$, as shown in the graph.
- a) Calculate the acceleration in each phase.
- b) In which phase is the acceleration greater in absolute value (without the sign)?
- c) Explain the meaning of the sign of each acceleration.
The acceleration of gravity: free fall
If you drop an eraser and a heavy book at the same time from the same height, they hit the floor almost together. Why, if the Earth pulls harder on the heavier object?
In free fall, with no air resistance, the speed of an object increases by $9.8\ \text{m/s}$ every second. What speed does a stone falling for $3\ \text{s}$ have? What is…
In free fall, with no air resistance, the speed of an object increases by $9.8\ \text{m/s}$ every second. What speed does a stone falling for $3\ \text{s}$ have? What is this acceleration called, and which letter is used to represent it?
A ball is dropped from a bridge, with negligible air resistance. Taking $g = 9.8\ \text{m/s}^2$, what speed does it have after $2\ \text{s}$ of falling? And after…
A ball is dropped from a bridge, with negligible air resistance. Taking $g = 9.8\ \text{m/s}^2$, what speed does it have after $2\ \text{s}$ of falling? And after $3.5\ \text{s}$?
A ballast bag is dropped from a stationary hot-air balloon, with negligible air resistance. a) What speed does it have after $2\ \text{s}$ of falling? (…
A ballast bag is dropped from a stationary hot-air balloon, with negligible air resistance.
- a) What speed does it have after $2\ \text{s}$ of falling? ($g = 9.8\ \text{m/s}^2$)
- b) And after $5\ \text{s}$?
- c) Your classmate says that, because it is heavy, the bag should fall with an acceleration greater than $9.8\ \text{m/s}^2$ even if there were no air. Are they right? Justify your answer.
Going fast is not the same as accelerating a lot
Back to the beginning: the bus pulling out of the stop has a nonzero acceleration while its velocity is changing; when it reaches a constant velocity, its acceleration is zero even if it is going very fast.
A car goes from $0$ to $20\ \text{m/s}$ in $10\ \text{s}$ and a motorbike goes from $0$ to $12\ \text{m/s}$ in $3\ \text{s}$. Calculate the acceleration of each vehicle.…
A car goes from $0$ to $20\ \text{m/s}$ in $10\ \text{s}$ and a motorbike goes from $0$ to $12\ \text{m/s}$ in $3\ \text{s}$. Calculate the acceleration of each vehicle. Which one accelerates more, even if the car ends up travelling faster?
A car pulls away at traffic lights: it accelerates from rest for $5\ \text{s}$ up to $15\ \text{m/s}$, maintains that speed for a while, and then brakes until it stops…
A car pulls away at traffic lights: it accelerates from rest for $5\ \text{s}$ up to $15\ \text{m/s}$, maintains that speed for a while, and then brakes until it stops in $3\ \text{s}$, as shown in the graph.
- a) Calculate the acceleration when pulling away.
- b) What is the acceleration while it maintains its speed? Why?
- c) Calculate the braking acceleration and compare it with the acceleration when pulling away.
Marcos states: "the car travelling faster is the one with greater acceleration". Analyse whether he is right using these data: car A goes from $0$ to $25\ \text{m/s}$ in…
Marcos states: "the car travelling faster is the one with greater acceleration". Analyse whether he is right using these data: car A goes from $0$ to $25\ \text{m/s}$ in $10\ \text{s}$ and car B goes from $0$ to $20\ \text{m/s}$ in $4\ \text{s}$, as shown in the graph.
- a) Calculate the acceleration of each car as it pulls away.
- b) Which one is going faster at the end? Which one has accelerated more?
- c) Explain in your own words the difference between "going fast" and "accelerating a lot".