Kinematics: Motion in One and Two Dimensions - Physics (Undergraduate Foundation)

Motion is the most fundamental concept in physics. This course provides the mathematical framework to describe it with quantitative precision. We will systematically analyse motion in one, two, and three dimensions, progressing from motion along a straight line to the vector analysis of projectiles and general spatial motion. This requires a full command of both vector algebra and introductory calculus. The principles of kinematics are the bedrock of physical analysis, engineering design, and biomechanics. This knowledge is required to analyse everything from vehicle performance and projectile trajectories to the movement of organisms and robotic motion. A command of this material is essential for any work involving the dynamics of moving systems, providing the tools to predict and control motion in the real world. Upon completion, you will be able to analyse motion in multiple dimensions. You will solve one-dimensional problems using the standard kinematic equations for constant acceleration. You will analyse two-dimensional projectile motion by resolving vectors into independent components. Critically, you will use vector calculus to describe the general case of motion with variable acceleration. This course is designed for first-year university students of the physical, biological, and medical sciences, as well as engineering and computer science. A firm command of vector algebra and introductory calculus (differentiation and integration) is a mandatory prerequisite. This material is the necessary foundation for the subsequent study of Newtonian dynamics and biomechanics.

1

Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[NUC Core] PHY 101: General Physics I - Mechanics
[NUC Core] PHY 101: General Physics I - Mechanics
This learning track provides a complete and rigorous treatment of introductory classical mechanics as specified by the NUC Core Curriculum. It is structured to build a comprehensive analytical framework, starting with the mathematical description of motion (kinematics) and progressing through its causes (Newtonian dynamics), the powerful conservation laws, the dynamics of rotating systems, and finally, the principles of universal gravitation. Mastery of this material is the non-negotiable foundation for all subsequent study in physics and engineering. The principles of classical mechanics are the operational language for analysing the physical world. This track provides the essential toolset for solving problems in every field of engineering, from aerospace to civil, and for understanding phenomena from the trajectory of a projectile to the orbits of planets. By the end of this track, you will be able to analyse motion using vectors and calculus, apply Newton's laws to solve any standard dynamics problem, use conservation laws to analyse complex systems and collisions, analyse rotational motion, and solve problems in celestial mechanics. This learning track is a mandatory prerequisite for all first-year university students of physics, engineering, and related physical sciences. It provides the foundational knowledge required for all subsequent courses in mechanics, electromagnetism, thermodynamics, and modern physics.

This learning track provides a complete and rigorous treatment of introductory classical mechanics as specified by the NUC Core Curriculum. It is structured to build a comprehensive analytical framework, starting with the mathematical description of motion (kinematics) and progressing through its causes (Newtonian dynamics), the powerful conservation laws, the dynamics of rotating systems, and finally, the principles of universal gravitation. Mastery of this material is the non-negotiable foundation for all subsequent study in physics and engineering. The principles of classical mechanics are the operational language for analysing the physical world. This track provides the essential toolset for solving problems in every field of engineering, from aerospace to civil, and for understanding phenomena from the trajectory of a projectile to the orbits of planets. By the end of this track, you will be able to analyse motion using vectors and calculus, apply Newton's laws to solve any standard dynamics problem, use conservation laws to analyse complex systems and collisions, analyse rotational motion, and solve problems in celestial mechanics. This learning track is a mandatory prerequisite for all first-year university students of physics, engineering, and related physical sciences. It provides the foundational knowledge required for all subsequent courses in mechanics, electromagnetism, thermodynamics, and modern physics.

Course Chapters

1. Introduction
5

This chapter establishes the rigorous language of kinematics. It defines the core quantities of motion – displacement, velocity, and acceleration – with a critical focus on the distinction between their scalar and vector forms. This foundation is the prerequisite for all subsequent analysis. Key objectives: distinguish between scalar and vector quantities; define average and instantaneous acceleration; and use differentiation to relate displacement, velocity, and acceleration.

Chapter lessons

1-1. Welcome
12:57

A direct statement of the course's purpose and structure. This lesson outlines the path from fundamental quantities and vectors to the full analysis of motion.

1-2. Position, distance and displacement
23:55

This lesson defines the foundational quantities of location and movement. We will establish an object's position, and critically distinguish between the total distance travelled (a scalar) and its overall displacement (a vector). This distinction is non-negotiable.

1-3. Speed and velocity (1)
11:42

This lesson defines the rate of change of position. We will distinguish between average speed, a scalar quantity derived from total distance, and average velocity, a vector quantity derived from displacement. This distinction is fundamental to all subsequent analysis.

1-4. Speed and velocity (2)
21:31

This lesson moves beyond averages to define instantaneous velocity – the rate of change of position at a specific moment. We will introduce this concept formally as the derivative of position with respect to time. Instantaneous speed is the magnitude of this resultant vector.

1-5. Acceleration
10:11

This lesson defines acceleration, the vector quantity describing the rate of change of velocity. We will establish both average and instantaneous acceleration, with the latter being the derivative of velocity with respect to time. An object accelerates if its speed or direction of motion changes.

2. Motion in One Dimension
4
3

This chapter applies kinematic principles to the simplest case: motion along a straight line. We will cover motion with variable acceleration using calculus, and then focus on the critical case of constant acceleration and its set of standard equations. Mastery of 1D motion is the required foundation for all further analysis. Key objectives: use calculus to relate displacement, velocity, and acceleration; apply the kinematic equations to solve constant acceleration problems; and analyse vertical motion under gravity.

Chapter lessons

2-1. Position, distance and displacement

This lesson formally defines position along a single coordinate axis. It establishes one-dimensional displacement as a vector quantity, reinforcing the critical distinction between this signed value and the total distance travelled.

2-2. Speed, velocity and acceleration

This lesson establishes the calculus-based relationships between the key kinematic quantities in one dimension. We formally define instantaneous velocity as the derivative of position, and acceleration as the derivative of velocity. Speed is the magnitude of the velocity.

2-3. Constant acceleration

This lesson focuses on the special case of constant acceleration. We will derive the standard set of algebraic kinematic equations that govern this type of motion. These equations are the essential tools for solving a wide class of mechanics problems.

2-4. Free-fall acceleration

This lesson examines motion under the sole influence of gravity. We define free-fall acceleration as a constant value near the Earth's surface. The standard kinematic equations are then directly applied to solve problems involving vertical motion.

3. Motion in Two or Three Dimensions
3
2

This chapter extends kinematic analysis to two and three dimensions. It establishes the general vector framework and calculus-based methods required to describe motion in a plane and in space. This mathematical toolkit is the direct prerequisite for analysing specific cases like projectile motion. Key objectives: describe position, velocity, and acceleration using standard vector notation; and apply differentiation and integration to these vectors to analyse motion with variable acceleration.

Chapter lessons

3-1. Position and displacement vectors

This lesson generalises the concept of position to two and three dimensions using the position vector. We define this vector by its components and establish the displacement vector as the change in an object's position.

3-2. Velocity vectors

This lesson defines the average velocity vector and the instantaneous velocity vector. The instantaneous velocity is always tangent to the particle's path, and its magnitude is the particle's speed.

3-3. Acceleration vectors

This lesson defines the average acceleration vector and instantaneous acceleration, the derivative of the velocity vector. Any change in a velocity vector's magnitude or its direction constitutes an acceleration.

4. Projectile Motion
2
1

This chapter provides a complete analysis of projectile motion, the primary application of two-dimensional kinematics. The core principle is the complete independence of horizontal motion (at constant velocity) and vertical motion (at constant acceleration). Key objectives: apply the principle of independent components to projectile problems; derive and use the equations for range, maximum height, and time of flight; and solve projectile trajectory problems.

Chapter lessons

4-1. Vertical and horizontal components

This lesson establishes the fundamental technique for analysing projectiles: separating the motion into two independent components. The horizontal component is treated as uniform motion, while the vertical component is treated as motion under constant gravitational acceleration.

4-2. Analysis of the trajectory

This lesson uses the principle of independent components and the standard kinematic equations to formally derive the formulae for a projectile's maximum height, time of flight, and horizontal range.

5. Uniform Circular Motion
2
1

This chapter examines uniform circular motion, a special case of two-dimensional motion where an object's speed is constant but its velocity is not. We will introduce the concept of an acceleration that serves only to change the direction of the velocity vector. Key objectives: define the descriptive quantities of circular motion (period, speed); explain the nature and direction of centripetal acceleration; and apply its formula to solve problems.

Chapter lessons

5-1. Motion parameters

This lesson defines the fundamental quantities used to describe uniform circular motion: the period, frequency, tangential speed, and angular velocity. These parameters form the basis for the subsequent kinematic analysis.

5-2. Centripetal acceleration

This lesson introduces centripetal acceleration, the acceleration required to maintain uniform circular motion. We establish that it is constant in magnitude and directed radially inward.

6. Relative Motion
1
1

This chapter introduces the concept of relative motion, establishing the framework for calculating kinematic quantities as measured by different observers. The analysis covers motion in different inertial frames of reference in both one and two dimensions. Key objectives: define an inertial frame of reference; apply the relative velocity formula in one dimension; and use vector subtraction to solve relative velocity problems in two dimensions.

Chapter lessons

6-1. Relative velocity

This lesson introduces the fundamental equation of relative velocity. We will establish the subscript notation, which provides a systematic framework for relating the velocity of an object as measured in different inertial frames of reference.

7. Conclusion
2

This chapter consolidates the core concepts of kinematics. It provides a structured summary of the mathematical tools and physical principles used to describe motion, reinforcing the foundation for the study of dynamics. The conclusion summarises the key definitions and equations. It also provides a forward look to the next course, where Newton's laws will be used to explain the causes of motion.

Chapter lessons

7-1. Summary

A concise review of vectors, the calculus of motion, and the kinematic equations. This lesson ensures all foundational material has been consolidated.

7-2. Newtonian dynamics

Explains how the principles of kinematics (the 'how' of motion) are a direct prerequisite for the study of dynamics (the 'why' of motion).