Mechanics Key Terms: A Comprehensive Glossary

Introduction

Mechanics involves the study of change in the motion of objects when acted upon by forces. A knowledge of mechanics enables an understanding of the behaviour of objects according to mathematical law. By applying these laws, we can model physical systems and predict the behaviour of objects that are under the influence of forces such as gravity and air resistance.

In HSC Mathematics Extension 2, precision in language is just as important as precision in algebra. This glossary provides the definitive mathematical definitions and illustrative examples for the core terminology used throughout the Mechanics syllabus.

Executive Summary

This guide serves as a quick-reference dictionary for:

Kinematics Terms: Displacement, velocity, speed, acceleration.
Dynamics Terms: Force, mass, Newton's Laws, gravity.
Specific Motion Types: Simple harmonic motion, resisted motion, projectile motion.
Vector Terminology: Magnitude, component, projection.

What is this about?

When an exam question asks for the "magnitude of the resistive force", or states that an object "oscillates with constant amplitude", you must know exactly what mathematical variable or formula that phrase translates to. Misinterpreting the difference between "velocity" (a vector with direction) and "speed" (a scalar) can lead to a completely incorrect integration. This page ensures your foundational vocabulary is flawless.

Main Content: Key Terms

A–D

Acceleration: The rate of change of velocity with respect to time ( $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ $a = \frac{d v}{d t} = \frac{d ^{2} x}{d t ^{2}}$ ). In mechanics, it can also be expressed as $v \frac{dv}{dx}$ $v \frac{d v}{d x}$ or $\frac{d}{dx}(\frac{1}{2}v^2)$ $\frac{d}{d x} (\frac{1}{2} v^{2})$ .
- Example: A car speeding up at $5 \text{ m/s}^2$ .
Amplitude: The maximum displacement of a particle from its centre of motion (equilibrium position) during an oscillation. Usually denoted by $A$ $A$ or $a$ $a$ .
- Example: A pendulum swinging $10 \text{ cm}$ left and right of the center has an amplitude of $10 \text{ cm}$ .
Component: A part of a vector broken down into a specific directional axis.
- Example: A force of $10\text{N}$ pushing at a $60^\circ$ angle has a horizontal component of $10\cos(60^\circ) = 5\text{N}$ .
Constant: A numerical value that does not change over time.
- Example: The acceleration due to gravity, $g \approx 9.8 \text{ ms}^{-2}$ , is a constant near the Earth's surface.
Displacement: A vector quantity representing the straight-line distance and direction of a particle from a defined origin (often denoted by $x$ $x$ or $\mathbf{r}$ $r$ ).
- Example: Walking $3\text{m}$ East and $4\text{m}$ West results in a displacement of $-1\text{m}$ (or $1\text{m}$ West), even though the total distance travelled is $7\text{m}$ .

F–M

Force: A vector quantity that causes a mass to accelerate, change direction, or deform. It is the product of mass and acceleration ( $F = ma$ $F = ma$ ).
- Example: A $10\text{ kg}$ block accelerating at $2\text{ m/s}^2$ requires a net force of $20\text{ Newtons}$ .
Gravity: The constant downward attractive force exerted by the Earth on a mass.
- Example: Gravity exerts a downward force of $mg$ on a projectile in flight.
Magnitude: The size or length of a vector quantity, ignoring its direction. It is always a non-negative scalar value.
- Example: If velocity is $-15 \text{ m/s}$ (moving downwards), the magnitude (speed) is just $15 \text{ m/s}$ .
Mass: A scalar measure of an object's resistance to acceleration (inertia) when a force is applied.
- Example: A bowling ball has a mass of $6\text{ kg}$ .

N–P

Newton’s Laws of Motion: Three fundamental physical laws mapping forces to acceleration.
- Example (2nd Law): Setting up the equation $m\ddot{x} = mg - kv$ for a falling object with air resistance.
Oscillate: To move back and forth continuously between two points around a central equilibrium position.
- Example: A mass bouncing up and down on a spring.
Period: The time taken for one complete cycle of an oscillation.
- Example: If an equation is $x = \cos(3t)$ , the period is $T = \frac{2\pi}{3}$ seconds.
Projectile: An object that is launched into the air and moves freely under the influence of gravity (and potentially air resistance).
- Example: A golf ball hit off a tee at an angle of $45^\circ$ .
Projection (Vectors): The orthogonal (perpendicular) "shadow" of one vector cast onto another.
- Example: The projection of gravity acting down an inclined plane is $mg \sin\theta$ .

R–T

Rectilinear: Motion constrained to a single straight line (one-dimensional motion).
- Example: An elevator moving purely up and down a shaft.
Resisted Motion: Motion where a particle moves through a medium that exerts a resistive force opposing the direction of motion.
- Example: A skydiver falling through the air.
Resistive Force: A force that opposes the motion of an object, usually proportional to a power of speed.
- Example: Air resistance $R = -kv^2$ pushing against a moving car.
Simple Harmonic Motion (SHM): A specific type of oscillatory rectilinear motion where the acceleration is directly proportional and opposite to displacement ( $\ddot{x} = -n^2 x$ $\overset{x}{¨} = - n^{2} x$ ).
- Example: The motion of a pendulum for small angles, governed by $\ddot{x} = -9x$ .
Speed: A scalar quantity representing the absolute magnitude of velocity ( $|v|$ $∣ v ∣$ ).
- Example: The speedometer in a car reads $60\text{ km/h}$ , regardless of whether you are driving North or South.
Terminal Velocity: The constant, maximum velocity reached by a falling object when upward resistance balances downward gravity (net force = $0$ $0$ ).
- Example: Setting $\ddot{x} = 0$ in the equation $m\ddot{x} = mg - kv^2$ to find the terminal velocity $v = \sqrt{\frac{mg}{k}}$ .
Trajectory: The curved geometric path followed by a projectile.
- Example: The parabolic curve traced by a thrown basketball.

V

Vector: A mathematical quantity that possesses both a magnitude (size) and a direction.
- Example: Velocity, force, and displacement are vectors. Mass and time are not.
Velocity: A vector quantity representing the rate of change of displacement with respect to time ( $v = \frac{dx}{dt}$ $v = \frac{d x}{d t}$ ).
- Example: A train moving at $80 \text{ km/h}$ due East.

mini-FAQ page

Q: Do I need to memorize these exact definitions for the exam? A: You will rarely be asked to write out a dictionary definition. However, you must know exactly what they mean to decode word problems. For example, if a question says "the particle reaches terminal velocity", you must instantly know to set acceleration ( $\ddot{x}$ ) to $0$ .

Q: What is the difference between displacement and trajectory? A: Displacement is a straight-line vector from the origin to your current position. Trajectory is the actual curved path your body took to get there.

Common mistakes to avoid

Confusing Velocity and Speed: Velocity can be negative (indicating direction, e.g., moving left or down). Speed is strictly the magnitude of velocity and must be positive. If you are integrating a resistive force proportional to speed squared ( $v^2$ ), the sign of $v$ requires extreme care depending on the direction of motion.
Mixing up Period and Frequency: Period is the time for one full cycle (seconds per cycle). Frequency (which occasionally appears) is the number of cycles per unit time (cycles per second). They are reciprocals of each other.

Practice on Vu's Maths Hub

Familiarity with Mechanics terminology is the first step to mastering the algebra.

Practice translating English word problems into differential equations in the HSC Mechanics Booklet.
Review the core concepts at Vu's Maths Hub.

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