How do I know which form of acceleration to use?

Look at the variable in the force function. If force is a function of time $t$, use $a = \frac{dv}{dt}$. If force is a function of displacement $x$, use $a = v\frac{dv}{dx}$. If force is a function of velocity $v$, you can use either, but the question will usually guide you towards finding either $v(t)$ or $v(x)$.

Can I ever use the suvat equations in this topic?

Only if you can prove that the net force on the particle is constant. If the force is given as a function like $F(t) = 2t$ or $F(x) = 3/x^2$, then the force is variable, acceleration is variable, and the suvat equations are not valid.

What's the difference between work done and the work-energy principle in these problems?

They are directly related. The work done by the net force on a particle equals the change in its kinetic energy. When you solve the equation $F(x) = mv\frac{dv}{dx}$ by integrating to get $\int F(x)\,dx = \int mv\,dv$, you are actually using the work-energy principle. The left side is the work done, and the right side integrates to $\frac{1}{2}mv^2$, representing kinetic energy.

Where can I learn Linear motion under a variable force for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 3.5 (Linear motion under a variable force) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/3-5-linear-motion-under-a-variable-force.

Is Linear motion under a variable force (3.5) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Linear motion under a variable force.

How do I revise Linear motion under a variable force for 9231 exams?

Start with the visual steps on this page, note the key exam points, then attempt a 9231 past-paper question on Linear motion under a variable force using MarkScheme's /mark tool. Mark strictly against the official scheme and re-read any marks you lost.

Linear motion under a variable force 9231 —…

Linear motion under a variable force

In your previous studies, you've likely spent a lot of time with the 'suvat' equations, which perfectly describe motion under constant acceleration. However, in the real world, forces are rarely constant. The thrust of a rocket changes as it burns fuel, air resistance increases with speed, and the gravitational pull of a planet varies with distance. This topic equips you with the calculus-based tools needed to analyse these more complex and realistic scenarios.

BY THE END, YOU CAN…

1Formulate a differential equation for the motion of a particle under a variable force.
2Use calculus to solve differential equations to find expressions for velocity and displacement.
3Solve problems where force is a function of time ( $F(t)$ ), displacement ( $F(x)$ ), or velocity ( $F(v)$ ).
4Apply initial or boundary conditions to find constants of integration and determine specific solutions.

In simple terms

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

Motion on the Move

This topic moves beyond the simple 'suvat' equations for constant acceleration. We use calculus to handle more realistic situations where the force on an object changes, causing its acceleration to vary.

THINK OF IT LIKE…

Imagine pushing a shopping trolley that gets heavier as you add items. You have to push harder to keep it accelerating. Your force isn't constant, so the trolley's acceleration isn't constant either. The simple 'suvat' equations assume the trolley's mass and your push are unchanging, which isn't the case here. We need a more powerful mathematical tool – calculus – to describe this variable motion.

1
Identify the variable force $F$ and use Newton's Second Law ( $F=ma$ ) to create an equation of motion.
2
Choose the correct form of acceleration: $\frac{dv}{dt}$ if force depends on time, or $v\frac{dv}{dx}$ if force depends on displacement.
3
Substitute the expression for acceleration into your equation to form a first-order differential equation.
4
Solve the differential equation by separating variables and integrating, using given conditions to find the constant of integration.

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.

$F_{net} = m \times a$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Newton's Second Law as a Differential Equation

The cornerstone of all dynamics problems is Newton's Second Law, $F=ma$ . When the force $F$ is not constant, the acceleration $a$ is also not constant. Since acceleration is the rate of change of velocity, and velocity is the rate of change of displacement, we are inherently dealing with derivatives. This means that the equation $F=ma$ becomes a differential equation.

$F_{net} = m \times a$

The key is to express the acceleration $a$ in a form that is useful for the given problem. This depends on whether the force is a function of time, displacement, or velocity.

The Three Forms of Acceleration

Choosing the correct expression for acceleration is the most critical first step in solving these problems. Your choice will determine the variables in your differential equation and how you go about solving it.

If force depends on time, $F(t)$ : Use $a = \frac{dv}{dt}$ . The equation of motion is $F(t) = m\frac{dv}{dt}$ . You can integrate with respect to $t$ to find $v(t)$ .
If force depends on displacement, $F(x)$ : Use $a = v\frac{dv}{dx}$ . The equation of motion is $F(x) = mv\frac{dv}{dx}$ . You can solve by separating variables to find a relationship between $v$ and $x$ .
If force depends on velocity, $F(v)$ : You have a choice. Use $a = \frac{dv}{dt}$ to find $v$ as a function of $t$ , or use $a = v\frac{dv}{dx}$ to find $v$ as a function of $x$ . The question will guide you.

Force Dependent on Displacement

When the force on a particle depends on its position, such as the force exerted by a spring or gravitational attraction, we must relate force to acceleration via displacement. This is where the $a = v \frac{dv}{dx}$ form is essential. This leads to a differential equation that can be solved by the method of separation of variables.

Worked examples

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

EXAMPLE 1

A particle P of mass 0.5 kg moves in a straight line. At time $t$ seconds, the net force acting on P is $(2t+1)$ N. When $t=0$ , the particle is at rest at the origin O. Find the velocity and displacement of P from O when $t=4$ s.

1
The force is a function of time, $F(t) = 2t+1$ . We use Newton's Second Law, $F=ma$ .

EXAMPLE 2

A particle of mass 2 kg is acted on by a force of magnitude $\frac{18}{x^2}$ N, directed away from the origin O. The particle is projected from the point $x=3$ with a velocity of 4 m/s towards O. Find its speed when it reaches the point $x=6$ .

1
The force is a function of displacement, $F(x) = \frac{18}{x^2}$ . The force is directed away from O, so it's in the positive $x$ direction.

How it all connects

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

MAIN IDEALinear motion under a variable force

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 / 11

What is the fundamental equation for motion under a variable force?

TAP OR SPACE TO FLIP

ANSWER

Newton's Second Law, $F = ma$ . Since the force $F$ is variable, the acceleration $a$ must also be variable.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
If force depends on time, $F(t)$ : Use $a = \frac{dv}{dt}$ . The equation of motion is $F(t) = m\frac{dv}{dt}$ . You can integrate with respect to $t$ to find $v(t)$ .
✓
If force depends on displacement, $F(x)$ : Use $a = v\frac{dv}{dx}$ . The equation of motion is $F(x) = mv\frac{dv}{dx}$ . You can solve by separating variables to find a relationship between $v$ and $x$ .
✓
If force depends on velocity, $F(v)$ : You have a choice. Use $a = \frac{dv}{dt}$ to find $v$ as a function of $t$ , or use $a = v\frac{dv}{dx}$ to find $v$ as a function of $x$ . The question will guide you.

Practice — then mark it

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

Real past paper[0]

Practice Questions

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 Questions 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

← PREVIOUS3.4 Hooke's law NEXT →3.6 Momentum

Linear motion under a variable force

BY THE END, YOU CAN…

1Formulate a differential equation for the motion of a particle under a variable force.
2Use calculus to solve differential equations to find expressions for velocity and displacement.
3Solve problems where force is a function of time ( $F(t)$ ), displacement ( $F(x)$ ), or velocity ( $F(v)$ ).
4Apply initial or boundary conditions to find constants of integration and determine specific solutions.

In simple terms

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

Motion on the Move

THINK OF IT LIKE…

1
Identify the variable force $F$ and use Newton's Second Law ( $F=ma$ ) to create an equation of motion.
2
Choose the correct form of acceleration: $\frac{dv}{dt}$ if force depends on time, or $v\frac{dv}{dx}$ if force depends on displacement.
3
Substitute the expression for acceleration into your equation to form a first-order differential equation.
4
Solve the differential equation by separating variables and integrating, using given conditions to find the constant of integration.

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.

$F_{net} = m \times a$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Newton's Second Law as a Differential Equation

$F_{net} = m \times a$

The key is to express the acceleration $a$ in a form that is useful for the given problem. This depends on whether the force is a function of time, displacement, or velocity.

The Three Forms of Acceleration

If force depends on time, $F(t)$ : Use $a = \frac{dv}{dt}$ . The equation of motion is $F(t) = m\frac{dv}{dt}$ . You can integrate with respect to $t$ to find $v(t)$ .
If force depends on displacement, $F(x)$ : Use $a = v\frac{dv}{dx}$ . The equation of motion is $F(x) = mv\frac{dv}{dx}$ . You can solve by separating variables to find a relationship between $v$ and $x$ .
If force depends on velocity, $F(v)$ : You have a choice. Use $a = \frac{dv}{dt}$ to find $v$ as a function of $t$ , or use $a = v\frac{dv}{dx}$ to find $v$ as a function of $x$ . The question will guide you.

Force Dependent on Displacement

Worked examples

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

EXAMPLE 1

1
The force is a function of time, $F(t) = 2t+1$ . We use Newton's Second Law, $F=ma$ .

EXAMPLE 2

1
The force is a function of displacement, $F(x) = \frac{18}{x^2}$ . The force is directed away from O, so it's in the positive $x$ direction.

How it all connects

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

MAIN IDEALinear motion under a variable force

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 / 11

What is the fundamental equation for motion under a variable force?

TAP OR SPACE TO FLIP

ANSWER

Newton's Second Law, $F = ma$ . Since the force $F$ is variable, the acceleration $a$ must also be variable.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
If force depends on time, $F(t)$ : Use $a = \frac{dv}{dt}$ . The equation of motion is $F(t) = m\frac{dv}{dt}$ . You can integrate with respect to $t$ to find $v(t)$ .
✓
If force depends on displacement, $F(x)$ : Use $a = v\frac{dv}{dx}$ . The equation of motion is $F(x) = mv\frac{dv}{dx}$ . You can solve by separating variables to find a relationship between $v$ and $x$ .
✓
If force depends on velocity, $F(v)$ : You have a choice. Use $a = \frac{dv}{dt}$ to find $v$ as a function of $t$ , or use $a = v\frac{dv}{dx}$ to find $v$ as a function of $x$ . The question will guide you.

Practice — then mark it

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

Real past paper[0]

Practice Questions

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 Questions 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

← PREVIOUS3.4 Hooke's law NEXT →3.6 Momentum

Revise Linear motion under a variable force for Cambridge 9231 (3.5)

In simple terms

Explore the concept

Key formulas

Full topic notes

Newton's Second Law as a Differential Equation

The Three Forms of Acceleration

Force Dependent on Displacement

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

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

Newton's Second Law as a Differential Equation

The Three Forms of Acceleration

Force Dependent on Displacement

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Linear motion under a variable force