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ESO Y3 · Physics and Chemistry
Interactions · Kinematics

Uniformly accelerated rectilinear motion

A car pulling away from traffic lights does not keep a constant speed: it gains speed little by little, always at the same rate. A falling stone does not fall at a constant speed either: it accelerates by the same amount every second. This motion, in a straight line and with constant acceleration, is UARM, and two equations describe it completely.

Before you start, make sure you know: Uniform rectilinear motion (2nd ESO)
Before you start, make sure you know: First-degree equations (2nd ESO)

What is UARM

If you film a falling ball or a scooter pulling away sharply, you will see that in the first second it covers a short stretch, in the next second a longer one and in the third one longer still. How can we describe exactly a motion in which the velocity changes at every instant but at a constant rate?

Animation of an accelerating moving object: an orange ball starts at the same time in the rows for 1 s, 2 s and 3 s with the same clock, and each row stops at its mark. The stretches between marks get longer and longer because the velocity increases. At the end everything fades out and starts again. each stretch is longer: faster and faster 0 s 1 s 2 s 3 s
Practice★☆☆
Explain in your own words what it means for a motion to be uniformly accelerated rectilinear motion (UARM) and give an example from everyday life.

Explain in your own words what it means for a motion to be uniformly accelerated rectilinear motion (UARM) and give an example from everyday life.

Solve
Practice★☆☆
State which of these motions are approximately a UARM and justify each answer: a) A car on a motorway with the speedometer fixed at $120\ \text{km/h}$. b) A ball falling…

State which of these motions are approximately a UARM and justify each answer:

  1. a) A car on a motorway with the speedometer fixed at $120\ \text{km/h}$.
  2. b) A ball falling from a roof.
  3. c) A lift starting up and gaining speed uniformly.
  4. d) A satellite orbiting the Earth.
Solve

The velocity equation

Move the sliders for initial velocity $v$ and acceleration $a$: notice that the $v(t)$ graph is now a sloping straight line (not horizontal, as in URM) and that the position $x(t)$ is a curve.

Practice★☆☆
A car starts from rest and accelerates at $2\ \text{m/s}^2$. Using equation $v = v_0 + a\cdot t$, calculate its velocity at $5\ \text{s}$.

A car starts from rest and accelerates at $2\ \text{m/s}^2$. Using equation $v = v_0 + a\cdot t$, calculate its velocity at $5\ \text{s}$.

Solve
Practice★☆☆
A cyclist already travelling at $5\ \text{m/s}$ accelerates at $3\ \text{m/s}^2$ for $4\ \text{s}$. What speed does the cyclist reach?

A cyclist already travelling at $5\ \text{m/s}$ accelerates at $3\ \text{m/s}^2$ for $4\ \text{s}$. What speed does the cyclist reach?

Solve
Practice★☆☆
A car travelling at $10\ \text{m/s}$ brakes with a deceleration of $-2\ \text{m/s}^2$. How long does it take to come to a complete stop?

A car travelling at $10\ \text{m/s}$ brakes with a deceleration of $-2\ \text{m/s}^2$. How long does it take to come to a complete stop?

Solve
Practice★★☆
An electric car goes from $8\ \text{m/s}$ to $23\ \text{m/s}$ in $5\ \text{s}$ with constant acceleration. What is its acceleration?

An electric car goes from $8\ \text{m/s}$ to $23\ \text{m/s}$ in $5\ \text{s}$ with constant acceleration. What is its acceleration?

Solve

Free fall and vertical projection

Ball thrown upwards from the ground with an initial velocity of 30 m/s (blue arrow pointing up) and the acceleration due to gravity of 10 m/s² directed towards the ground (green arrow pointing down). v₀ = 30 m/s g = 10 m/s² towards the ground upwards
Practice★☆☆
A stone is dropped from a viewpoint. Taking $g = 10\ \text{m/s}^2$ so that the numbers are neat, what is its speed after $4\ \text{s}$ of falling?

A stone is dropped from a viewpoint. Taking $g = 10\ \text{m/s}^2$ so that the numbers are neat, what is its speed after $4\ \text{s}$ of falling?

Solve
Practice★☆☆
You throw a ball vertically upwards at $30\ \text{m/s}$, as shown in the figure. With $g = 10\ \text{m/s}^2$, its speed decreases by $10\ \text{m/s}$ every second. What…

You throw a ball vertically upwards at $30\ \text{m/s}$, as shown in the figure. With $g = 10\ \text{m/s}^2$, its speed decreases by $10\ \text{m/s}$ every second.

v₀ = 30 m/s g = 10 m/s² t = 1 s: v = ? t = 2 s: v = ?

What speed does it have $1\ \text{s}$ after launch? And after $2\ \text{s}$?

Solve
Practice★★☆
You throw a ball vertically upwards at $20\ \text{m/s}$ from the ground ($g = 10\ \text{m/s}^2$). a) How long does it take to reach maximum height, which is when its…

You throw a ball vertically upwards at $20\ \text{m/s}$ from the ground ($g = 10\ \text{m/s}^2$).

  1. a) How long does it take to reach maximum height, which is when its velocity becomes zero?
  2. b) Using $x = v_0\cdot t - \frac{1}{2}g\cdot t^2$, calculate that maximum height.
Solve

The position equation

A car starts from rest with $a = 2\ \text{m/s}^2$: in $1\ \text{s}$ it covers $1\ \text{m}$, in $2\ \text{s}$ it has covered $4\ \text{m}$ and in $3\ \text{s}$ it is already at $9\ \text{m}$. Why does the distance travelled not increase in proportion to the time elapsed, and which formula lets us calculate it?

Position–time graph of UARM starting from rest: a parabola that starts horizontally at the origin and rises faster and faster, with points at 1 s, 2 s and 3 s (1 m, 4 m and 9 m). t (s) x (m) 1 2 3 0 1 m 4 m 9 m x(t): curve, not straight x = ½ a t²
Practice★★☆
A flowerpot falls from a height of $45\ \text{m}$, as shown in the figure. Taking $g = 10\ \text{m/s}^2$ and knowing that in free fall the distance travelled is…

A flowerpot falls from a height of $45\ \text{m}$, as shown in the figure. Taking $g = 10\ \text{m/s}^2$ and knowing that in free fall the distance travelled is $x = \frac{1}{2}g\cdot t^2$:

45 m g = 10 m/s² t = ? v = ?
  1. a) How long does it take to hit the ground?
  2. b) At what speed does it hit the ground?
Solve
Practice★★☆
A car starts from rest with an acceleration of $3\ \text{m/s}^2$ for $4\ \text{s}$. Calculate the distance it travels using $x = \frac{1}{2}a\cdot t^2$ and its speed at…

A car starts from rest with an acceleration of $3\ \text{m/s}^2$ for $4\ \text{s}$. Calculate the distance it travels using $x = \frac{1}{2}a\cdot t^2$ and its speed at the end of those $4\ \text{s}$.

Solve
Practice★★★
A car travelling at $10\ \text{m/s}$ accelerates at $2\ \text{m/s}^2$ for $5\ \text{s}$, as shown in the figure. a) What speed does it reach? b) What distance does it…

A car travelling at $10\ \text{m/s}$ accelerates at $2\ \text{m/s}^2$ for $5\ \text{s}$, as shown in the figure.

v₀ = 10 m/s a = 2 m/s²
  1. a) What speed does it reach?
  2. b) What distance does it travel in those $5\ \text{s}$? Use $x = v_0\cdot t + \frac{1}{2}a\cdot t^2$.
  3. c) Why is that distance not simply $10\ \text{m/s}$ multiplied by $5\ \text{s}$? Explain why.
Solve

Summary

Let's practice! 8 medium exercises in this section
★★☆ An electric car goes from $8\ \text{m/s}$ to $23\ \text{m/s}$ in $5\ \text{s}$ with…★★☆ After $4\ \text{s}$ accelerating at $3\ \text{m/s}^2$, an electric scooter reaches…★★☆ A train accelerates from $5\ \text{m/s}$ with an acceleration of $2,5\ \text{m/s}^2$. How…
Let's practice! 8 hard exercises in this section
★★★ A toy rocket ascends with an initial velocity of $30\ \text{m/s}$ and, once the engine…★★★ From a balcony $10\ \text{m}$ above the ground, you throw a ball vertically downwards at…
See all the exercises in this section