What is the difference between speed and velocity?

Speed is a scalar quantity—it only has magnitude (e.g., 20 m s⁻¹). Velocity is a vector quantity—it has both magnitude and direction (e.g., 20 m s⁻¹ due north). In straight-line motion, we often use positive and negative signs to indicate direction.

When should I use g = 9.8 m s⁻² and when should I use g = 10 m s⁻²?

The Cambridge International exam papers will always specify the value of 'g' to use on the front cover of the question paper. For P4 Mechanics, this is typically g = 10 m s⁻². However, always check the instructions in your specific exam. If you are doing practice problems from other sources, they may use g = 9.8 or 9.81.

How do I choose the right SUVAT equation?

First, write down the values you know for s, u, v, a, and t. Identify the variable you want to find. Then, look for the equation that contains the four variables you are working with (the knowns and the one unknown). Each of the five equations omits exactly one of the five variables.

What does it mean if I calculate a negative displacement or velocity?

A negative sign simply indicates direction. In a straight-line motion problem, you must first define a positive direction (e.g., upwards, or to the right). A negative displacement then means the object is on the opposite side of the origin from the positive direction. A negative velocity means the object is moving in the negative direction.

Where can I learn Kinematics of motion in a straight line for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 4.2 (Kinematics of motion in a straight line) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/4-2-kinematics-of-motion-in-a-straight-line.

Is Kinematics of motion in a straight line (4.2) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Kinematics of motion in a straight line.

Kinematics of motion in a straight line 9709…

Kinematics of motion in a straight line

Welcome to the study of kinematics, the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this lesson, we will develop the mathematical tools needed to analyse motion in a straight line, a fundamental skill for all further mechanics topics.

BY THE END, YOU CAN…

1Apply the five constant acceleration (SUVAT) equations to solve problems.
2Interpret displacement-time and velocity-time graphs to find displacement, velocity, and acceleration.
3Use differentiation and integration to model motion with variable acceleration.
4Understand and apply the concept of acceleration due to gravity for objects in free fall.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Language of Motion

Kinematics provides the tools to describe and predict how things move. We use equations, graphs, and calculus to understand an object's position, speed, and acceleration.

THINK OF IT LIKE…

Imagine you're driving a car. Pressing the accelerator gives you a constant acceleration, and we can use 'SUVAT' equations to predict your speed and distance travelled after a certain time. The car's journey can also be drawn as a graph, where a steepening line shows you're speeding up, and a flat line shows you're cruising. Calculus is like having a super-powered speedometer that works even when your acceleration isn't constant.

1
SUVAT for constant acceleration — pick the equation that doesn't involve the variable you don't need.
2
The gradient of a displacement-time graph is velocity. The gradient of a velocity-time graph is acceleration.
3
For objects in free fall, acceleration 'a' is the acceleration due to gravity, 'g' (approx 9.8 m s⁻²). Remember to define a positive direction and stick to it.
4
When acceleration changes, use calculus: differentiate displacement to get velocity (v = ds/dt), and differentiate velocity to get acceleration (a = dv/dt).

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$v = \frac{ds}{dt} \quad \implies \quad s = \int v \, dt$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Motion with Constant Acceleration: The SUVAT Equations

When an object moves with constant, uniform acceleration, its motion can be described by a set of five equations. These are known as the SUVAT equations, named after the variables they relate: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Mastering these is essential for solving a wide range of mechanics problems.

$v = u + at$
$s = ut + \frac{1}{2}at^2$
$s = vt - \frac{1}{2}at^2$
$s = \frac{1}{2}(u+v)t$
$v^2 = u^2 + 2as$

EXAM TIP

Before applying a SUVAT equation, always list the variables you know and the one you need to find. This helps you select the correct equation. Remember, these equations are only valid if acceleration is constant.

Graphical Representation of Motion

Graphs are a powerful visual tool in kinematics. A displacement-time graph plots the object's displacement from a starting point against time. A velocity-time graph plots the object's velocity against time. The features of these graphs have important physical meanings.

Displacement-Time Graph (s-t): The gradient at any point gives the instantaneous velocity.
Velocity-Time Graph (v-t): The gradient at any point gives the instantaneous acceleration.
Velocity-Time Graph (v-t): The area between the graph line and the time axis gives the displacement.

Motion with Variable Acceleration

When acceleration is not constant (for example, it might depend on time), the SUVAT equations cannot be used. In these cases, we must use calculus. The fundamental relationships are based on the idea that velocity is the rate of change of displacement, and acceleration is the rate of change of velocity.

Velocity: $v = \frac{ds}{dt} \quad \implies \quad s = \int v \, dt$

Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \quad \implies \quad v = \int a \, dt$

EXAM TIP

When integrating to find velocity or displacement, do not forget the constant of integration, '+ C'. Use the initial conditions given in the problem (e.g., 'starts from rest at the origin' means v=0 and s=0 when t=0) to find the value of C.

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A car is travelling along a straight road at 12 m s⁻¹. It accelerates uniformly at 1.5 m s⁻² for 8 seconds. Find the car's final velocity and the distance it travels during this time.

1
First, list the known SUVAT variables. We are given: $u = 12 \text{ m s}^{-1}$ $a = 1.5 \text{ m s}^{-2}$ $t = 8 \text{ s}$

EXAMPLE 2

The velocity-time graph for a cyclist's journey is a straight line from (0, 4) to (10, 12), where time is in seconds and velocity is in m s⁻¹. Calculate the acceleration of the cyclist and the total distance travelled in the 10 seconds.

1
The graph is a straight line, which means the acceleration is constant.

EXAMPLE 3

A particle moves in a straight line. Its acceleration at time $t$ seconds is given by $a = (6t - 2)$ m s⁻². When $t=0$ , the particle is at the origin and has a velocity of 3 m s⁻¹. Find expressions for its velocity and displacement in terms of $t$ .

1
We are given $a = 6t - 2$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAKinematics of motion in a straight line

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What do the letters in SUVAT stand for?

TAP OR SPACE TO FLIP

ANSWER

s: displacement, u: initial velocity, v: final velocity, a: acceleration, t: time.

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
$v = u + at$
✓
$s = ut + \frac{1}{2}at^2$
✓
$s = vt - \frac{1}{2}at^2$
✓
$s = \frac{1}{2}(u+v)t$
✓
$v^2 = u^2 + 2as$

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Kinematics Problems

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Kinematics Problems on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS4.1 Forces and equilibrium NEXT →4.3 Momentum

Kinematics of motion in a straight line

BY THE END, YOU CAN…

1Apply the five constant acceleration (SUVAT) equations to solve problems.
2Interpret displacement-time and velocity-time graphs to find displacement, velocity, and acceleration.
3Use differentiation and integration to model motion with variable acceleration.
4Understand and apply the concept of acceleration due to gravity for objects in free fall.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Language of Motion

Kinematics provides the tools to describe and predict how things move. We use equations, graphs, and calculus to understand an object's position, speed, and acceleration.

THINK OF IT LIKE…

1
SUVAT for constant acceleration — pick the equation that doesn't involve the variable you don't need.
2
The gradient of a displacement-time graph is velocity. The gradient of a velocity-time graph is acceleration.
3
For objects in free fall, acceleration 'a' is the acceleration due to gravity, 'g' (approx 9.8 m s⁻²). Remember to define a positive direction and stick to it.
4
When acceleration changes, use calculus: differentiate displacement to get velocity (v = ds/dt), and differentiate velocity to get acceleration (a = dv/dt).

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$v = \frac{ds}{dt} \quad \implies \quad s = \int v \, dt$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Motion with Constant Acceleration: The SUVAT Equations

$v = u + at$
$s = ut + \frac{1}{2}at^2$
$s = vt - \frac{1}{2}at^2$
$s = \frac{1}{2}(u+v)t$
$v^2 = u^2 + 2as$

EXAM TIP

Graphical Representation of Motion

Displacement-Time Graph (s-t): The gradient at any point gives the instantaneous velocity.
Velocity-Time Graph (v-t): The gradient at any point gives the instantaneous acceleration.
Velocity-Time Graph (v-t): The area between the graph line and the time axis gives the displacement.

Motion with Variable Acceleration

Velocity: $v = \frac{ds}{dt} \quad \implies \quad s = \int v \, dt$

Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \quad \implies \quad v = \int a \, dt$

EXAM TIP

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A car is travelling along a straight road at 12 m s⁻¹. It accelerates uniformly at 1.5 m s⁻² for 8 seconds. Find the car's final velocity and the distance it travels during this time.

1
First, list the known SUVAT variables. We are given: $u = 12 \text{ m s}^{-1}$ $a = 1.5 \text{ m s}^{-2}$ $t = 8 \text{ s}$

EXAMPLE 2

1
The graph is a straight line, which means the acceleration is constant.

EXAMPLE 3

1
We are given $a = 6t - 2$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAKinematics of motion in a straight line

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What do the letters in SUVAT stand for?

TAP OR SPACE TO FLIP

ANSWER

s: displacement, u: initial velocity, v: final velocity, a: acceleration, t: time.

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
$v = u + at$
✓
$s = ut + \frac{1}{2}at^2$
✓
$s = vt - \frac{1}{2}at^2$
✓
$s = \frac{1}{2}(u+v)t$
✓
$v^2 = u^2 + 2as$

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Kinematics Problems

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Kinematics Problems on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS4.1 Forces and equilibrium NEXT →4.3 Momentum

Revise Kinematics of motion in a straight line for Cambridge 9709 (4.2)

In simple terms

Explore the concept

Key formulas

Full topic notes

Motion with Constant Acceleration: The SUVAT Equations

Graphical Representation of Motion

Motion with Variable Acceleration

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Kinematics Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Motion with Constant Acceleration: The SUVAT Equations

Graphical Representation of Motion

Motion with Variable Acceleration

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Kinematics Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Kinematics of motion in a straight line