Acceleration
A car pulls away from a traffic light on the Autopista Duarte: it doesn't keep the same speed all the time, it gains a little more every second. That quantity, how fast velocity changes, is 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 can we measure, with a single quantity, the rate at which that velocity changes?
Go back to the motorcycle from the beginning: it gains $4\ \text{m/s}$ every second, so its acceleration is $4\ \text{m/s}^2$.
The speedometer of the train you are travelling on shows an increasing speed as it leaves the station. Explain in your own words what acceleration is and which two…
The speedometer of the train you are travelling on shows an increasing speed as it leaves the station. Explain in your own words what acceleration is and which two physical quantities it relates to time.
A skater starts from rest and reaches $10\ \text{m/s}$ in $5\ \text{s}$. Calculate their acceleration using $a = \frac{\Delta v}{\Delta t}$ and state its unit.
A skater starts from rest and reaches $10\ \text{m/s}$ in $5\ \text{s}$. Calculate their acceleration using $a = \frac{\Delta v}{\Delta t}$ and state its unit.
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
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.
Two cars carry out an emergency stop from $30\ \text{m/s}$: the first comes to a stop in $6\ \text{s}$ and the second in $4\ \text{s}$, as shown in the graph. Calculate…
Two cars carry out an emergency stop from $30\ \text{m/s}$: the first comes to a stop in $6\ \text{s}$ and the second in $4\ \text{s}$, as shown in the graph.
Calculate the acceleration of each car and explain which of the two brakes more intensely.
The acceleration of gravity: free fall
If you drop an eraser and a heavy book at the same time from the same height, both hit the floor together. Why, if the Earth pulls harder on the heavier object?
Throwing upward: the vertical throw
If you throw a ball upward, gravity acts the same way: it always pulls downward. Taking the upward direction as positive, the acceleration is $a = -g = -9.8\ \text{m/s}^2$ throughout the whole trip, both going up and coming down. With the same equation $v = v_0 + a \cdot t$ we get:
$$v = v_0 - g \cdot t$$On the way up, the velocity (positive) and the acceleration (negative) have opposite signs: the ball slows down. Each second it loses $9.8\ \text{m/s}$. At the highest point $v = 0$, and then the velocity becomes negative: it comes down faster and faster.
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 $3.5\ \text{s}$?
From the top of a cliff, a $100\ \text{g}$ stone and a $2\ \text{kg}$ stone are dropped at the same time, with negligible air resistance, as shown in the figure. a) What…
From the top of a cliff, a $100\ \text{g}$ stone and a $2\ \text{kg}$ stone are dropped at the same time, with negligible air resistance, as shown in the figure.
- a) What speed does each stone have after $3\ \text{s}$ of falling? ($g = 9.8\ \text{m/s}^2$)
- b) Which of the two reaches the ground first?
- c) Explain why mass does not appear in these calculations.
Going fast is not the same as accelerating a lot
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".