Linear Motion Under a Variable Force

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Solve problems which can be modelled as the linear motion of a particle under the action of a variable force, by setting up and solving an appropriate differential equation
    • Including use of v dv/dx for acceleration, where appropriate
    • Only differential equations in which the variables are separable are included

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What is a Variable Force?

A variable force is a force that changes in value. Unlike constant forces (such as the weight of an object, which stays the same), variable forces can depend on:

  • Time (t) — the force gets stronger or weaker as time passes
  • Position (x) — the force changes depending on where the particle is
  • Velocity (v) — the force depends on how fast the particle is moving

Example: A spring force F = -kx depends on position. Air resistance F = -kv² depends on velocity.

Newton's Second Law with Variable Forces

Newton's second law tells us that:

F = ma

where:

  • F = force acting on the particle (in newtons, N)
  • m = mass of the particle (in kg)
  • a = acceleration of the particle (in m s⁻²)

When the force varies, we write this as a differential equation (an equation involving rates of change).

Different Forms of Acceleration

Acceleration can be written in different ways depending on the problem:

1. a = dv/dt (acceleration as the rate of change of velocity with respect to time)

  • Use this when the force depends on time or velocity

2. a = v dv/dx (acceleration in terms of velocity and position)

  • Use this when the force depends on position (x)
  • This form comes from the chain rule: a = dv/dt = (dv/dx)(dx/dt) = v(dv/dx)

When to use which:

  • If F depends on x → use a = v dv/dx
  • If F depends on t or v → use a = dv/dt

Setting Up the Differential Equation

Step 1: Identify the force acting on the particle and what it depends on.

Step 2: Apply Newton's second law: F = ma

Step 3: Choose the correct form of acceleration:

  • If F = F(x), write F = m v dv/dx
  • If F = F(t) or F = F(v), write F = m dv/dt

Step 4: Rearrange to get a separable differential equation (one where you can get all terms involving one variable on one side, and all terms involving the other variable on the other side).

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