Linear Motion under a variable force

Essentials of Linear Motion under a Variable Force

The force acting on an object is variable, rather than constant as in basic mechanics. When force changes over time, it alters object’s velocity and position.
The force is often expressed as a function of time, position, or velocity. This complicates the force-equation but also adds greater detail and flexibility to the problem.
Newton’s Second Law of Motion still applies: F = ma. Here however, a, the acceleration, is the derivative of the velocity which again is the derivative of position. Since the force is also changing, these properties interlock in new ways which require differential equations to solve.

Understanding Variable Forces

Drag Forces: These act in the opposite direction to the motion of the particle. They can depend on speed, the square of speed, or be constant.
Spring Forces: According to Hooke’s law, these are proportional to the extension or compression of a spring. The force is given as F = -kx where k is the spring constant and x the position.
Gravitational Forces: Act vertically downwards. These can become variable in problems involving extremely great heights (as in problems involving spacecraft), where the gravitational field strength diminishes with altitude.

Key Equations

The definition of force and acceleration holds firm: F = ma. Both sides of this equation can be variable.
For drag forces: F = -kv or F = -k(v^2) where k is the drag constant, and v is velocity.
For spring forces: F = -kx where x is the displacement from the rest or equilibrium position and k is the spring constant.
The gravitational force is F = mg where m is mass and g is the gravitational field strength. However, for large altitudes (as in space mechanics) it becomes F = GMm/r² with M the mass of Earth, m the mass of the particle, r the distance from the centre of Earth, and G the gravitational constant.

Analyzing Problems with Variable Forces

You generally need calculus to solve these problems, specifically differential equations. These allow for the study of how one quantity changes in relation to another, opening up the mechanics to variables changing over time.
The wording in the problem can often indicate which type of force you’re being asked to consider. Look for words like “drag”, “resistance”, “spring”, or “gravity”.
Sketching force diagrams is still very useful. Being able to visualize how the forces at work often leads to clearer understanding of the problem.
The Analytic approach requires step-by-step integration, and often gives a solution in terms of an indefinite integral.
The Numeric approach uses numerical approximations, particularly useful where an analytic approach is challenging or impossible. Examples include problems involving air resistance proportional to velocity squared.