QCE Specialist Mathematics - Unit 4 - Modelling motion

Modelling Motion and Simple Harmonic Motion | QCE Specialist Mathematics

Learn QCE Specialist motion modelling, forces, momentum, acceleration forms and simple harmonic motion.

Updated 2026-05-18 - 5 min read

Modelling motion brings calculus and mechanics together. Instead of only tracking displacement, velocity and acceleration, you now connect motion to forces.

Momentum and force

Momentum is:

$ p=mv. $

Momentum has units $\text{kg m s}^{-1}$. Force has units newtons, where:

$ 1\text{ N}=1\text{ kg m s}^{-2}. $

For constant mass, force is the rate of change of momentum:

$ F=\frac{dp}{dt}=m\frac{dv}{dt}=ma. $

If several forces act on the particle, use the resultant force:

$ \sum F=ma. $

Action and reaction forces come in pairs on different bodies. They are equal in magnitude and opposite in direction, but they do not cancel when you are drawing a force diagram for one body. Only forces acting on the chosen body belong in that body's equation of motion.

Typical forces include weight $mg$, normal reaction $R$, tension $T$, applied force, friction and air resistance. Friction and air resistance oppose motion or attempted motion; they are not Newton's third-law reaction forces for the body being modelled. The normal reaction is perpendicular to the contact surface. Tension acts along the string or cable, away from the particle.

Resolving forces

For inclined planes and concurrent force problems, it is often easier to choose axes parallel and perpendicular to the plane. Then weight is the force that needs to be resolved into components.

On an incline of angle $\theta$, weight $mg$ has components:

$ mg\sin\theta $

down the plane and

$ mg\cos\theta $

into the plane.

The normal reaction balances the perpendicular component when there is no acceleration away from the plane.

For a smooth inclined plane, friction is ignored. The force down the plane is usually $mg\sin\theta$, and the perpendicular balance is usually $R=mg\cos\theta$ if there are no other perpendicular forces. Connected bodies and pulleys are excluded from this syllabus point, so keep the model to a single body under concurrent forces.

If the plane is rough, friction has magnitude:

$ F_f=\mu R $

when limiting friction is being used, or $F_f\le\mu R$ when the body is at rest and friction adjusts as needed. A common setup is a block on an incline, with axes chosen parallel and perpendicular to the plane:

$ \sum F_{\parallel}=ma,\quad \sum F_{\perp}=0. $

If the body is moving at constant velocity or remains at rest, then $a=0$ along the plane. That does not mean there are no forces. It means the forces balance in the chosen direction.

For a rough incline with a pulling force $P$ up the plane and motion up the plane, the parallel equation might be:

$ P-mg\sin\theta-\mu R=ma. $

The perpendicular equation is usually:

$ R=mg\cos\theta. $

The signs would change if the positive direction or motion direction changed.

Acceleration forms

In one-dimensional motion, acceleration can be written in several useful ways:

$ a=\frac{dv}{dt}=\frac{d^2x}{dt^2}=v\frac{dv}{dx}=\frac{d}{dx}\left(\frac12v^2\right). $

If acceleration is given in terms of time, use $\frac{dv}{dt}$. If it is given in terms of displacement, $v\frac{dv}{dx}$ or $\frac{d}{dx}\left(\frac12v^2\right)$ is usually cleaner.

For non-constant force, Newton's second law becomes a differential equation. If $F$ is given as a function of displacement, then:

$ m v\frac{dv}{dx}=F(x). $

This is why the alternative acceleration forms matter; they let the calculus match the variable used in the force law.

For vertical motion with resistance proportional to $v^2$, choose the positive direction before writing the force equation. If upward is positive and the particle is moving upward, gravity and air resistance both act downward. A model might look like:

$ m\frac{dv}{dt}=-mg-kv^2. $

If the force depends on position, use:

$ a=v\frac{dv}{dx}. $

For example, with upward positive:

$ mv\frac{dv}{dx}=-mg-7mv^2. $

Separating this equation can find height as a function of speed. The maximum height occurs when $v=0$, not when acceleration is $0$.

Simple harmonic motion

Simple harmonic motion occurs when acceleration is proportional to displacement from the centre, but directed back toward the centre:

$ a=-\omega^2x. $

The standard solutions are:

$ x=A\sin(\omega t+\alpha) $

or

$ x=A\cos(\omega t+\beta). $

Here $A$ is the amplitude and $\omega$ is the angular frequency. The period is:

$ T=\frac{2\pi}{\omega}. $

The velocity-displacement relationship is:

$ v^2=\omega^2(A^2-x^2). $

The frequency is the reciprocal of the period:

$ f=\frac1T. $

In SHM, $x$ is measured from the equilibrium position. If the centre of motion is $x=h$, rewrite the displacement from equilibrium as $x-h$ before applying the standard formulas.

The velocity relation follows from:

$ a=v\frac{dv}{dx}=-\omega^2x. $

So:

$ v\,dv=-\omega^2x\,dx. $

Integrating:

$ \frac12v^2=-\frac12\omega^2x^2+c. $

At an endpoint, $x=A$ and $v=0$, so $c=\frac12\omega^2A^2$. Therefore:

$ v^2=\omega^2(A^2-x^2). $

This shows the key physical facts: speed is greatest at the centre, zero at the endpoints, and acceleration has greatest magnitude at the endpoints. If the centre is $x=h$, use:

$ a=-\omega^2(x-h) $

and the endpoints are $h-A$ and $h+A$.

The phase in $x=A\sin(\omega t+\alpha)$ or $x=A\cos(\omega t+\beta)$ is chosen from the initial position and velocity. It is not decoration; it tells you where in the oscillation the particle starts.

Vertical motion and resistance

Vertical motion under gravity uses weight $mg$ and a chosen positive direction. Without air resistance, acceleration is constant, usually $-g$ if upward is positive. With air resistance, the resistance force acts opposite the velocity, so the sign can change depending on whether the body is moving up or down.

The safest approach is to draw the forces at the instant described and then write $\sum F=ma$. After that, choose the acceleration form that matches the variable in the equation.

Quick check

Sources

• QCAA Specialist Mathematics subject page
• QCAA Specialist Mathematics 2025 syllabus
• OpenStax University Physics Volume 1: Newton's Second Law
• OpenStax University Physics Volume 1: Simple Harmonic Motion