If you are sitting in a chair reading this, are you moving? Common sense tells you NO. Your chair is planted firmly on the floor, your coffee mug isn't sliding, and your room looks blissfully stationary.
Yet, physics tells a completely different story [1]. Motion is not an absolute state that an object simply "owns." It is a dynamic relationship between objects and observers. It depends on who looks at it.
Imagine you are sitting on a bus that is smoothly driving, next to a cardboard box on the seat beside you. Meanwhile, a friend is standing still on the sidewalk watching the bus drive past.
Let us ask: Is the box moving?
The short answer is: It depends on who you ask.
Your Answer: "No, the box is stationary."
Your friend's Answer: "Yes, the box is moving."
Because the answer depends on whom you are asking, movement is considered in physics to be relative.
The Cosmic Perspective: Scaling Up Your Hidden Speeds
Let’s expand your perspective across expanding scales of space. Relative to your room, your speed is $0 \text{ m/s}$. However, because Earth is spinning on its axis, you are rotating at roughly $1,600 \text{ km/h}$ if you live near the equator.
Zoom out further: Earth orbits the Sun at an astounding average speed of $107,000 \text{ km/h}$. And your entire solar system is revolving around the centre of the Milky Way galaxy at roughly $800,000 \text{ km/h!}$.
The core thesis of kinematics is clear: there is no absolute point labelled "Rest" or "Moving". Let's see how this builds from answering a question as simple as "where are things located at?"
The Universal Clock ($dt' = dt$)
Before we map space, classical mechanics must establish a background parameter for time [3]. According to Newton, time acts like an impartial, universal metronome ticking identically for everyone. This assumption is very commonsense and is the foundation of the whole theory.
Axiom 0 (Absolute Time): Time flows identically for all observers regardless of their state of motion or location $(\Delta t_A = \Delta t_B = \Delta t$) [3].
This means that whether you are standing still or speeding through space on a supersonic jet, $\text{1 second}$ elapsed on your watch equals a second elapsed on a friend's watch. In mathematical shorthand, time interval transformations reduce to $dt' = dt$ [3].
Position is Relative (Stationary Frame Shifts)
With a universal clock established, we can introduce spatial coordinates [2]. To locate any object, you must first declare a reference frame complete with an origin point $O(0,0,0)$ and coordinate axes $(x, y, z)$, measured from the Origin.
Axiom 1( Spatial Homogeneity): Space is homogeneous: it has no preferred centre or absolute origin [3]. Wherever you choose to place your origin point is entirely up to you.
Shifted Frames ($\mathbf{r}' = \mathbf{r} - \mathbf{r}_0$) and Invariant Distance
If Observer A places their origin at the corner flag of a soccer pitch and Observer B places theirs at the centre circle, their position vectors to the ball will differ by a constant offset vector $\mathbf{r}_0$ [3]:
$$\mathbf{r}' = \mathbf{r} - \mathbf{r}_0$$
An illustration is shown in the figure where $O$ and $O'$ are different reference frames.
The straight-line distance between any two objects.
The arrow (direction and length) pointing from location A to location B.
Layer 2: Velocity is Relative (Inertial Frames)
When position changes continuously with time, we define velocity as $\mathbf{v} = \frac{d\mathbf{r}}{dt}$ [3]. But what happens if your reference frame ($O'$) itself is moving at a constant velocity $\mathbf{u}$?
The Galilean Transformation $(\mathbf{r}' = \mathbf{r} - \mathbf{u}t)$
If $O'$ starts moving with a speed $u$, $\mathbf{r}_0$ is not a constant but changing with time according to,
$$\mathbf{r}_0 = ut$$
Reference frames moving at a constant velocity are known as inertial frames [3]. Galileo recognised that spatial coordinates transform linearly under constant motion:
$$\mathbf{r}' = \mathbf{r} - \mathbf{u}t$$
Walking Inside a Moving Train: Relative Velocity in Action
Imagine walking toward the front of a train carriage at $5 \text{ km/h}$ while the train chugs down the tracks at $100 \text{ km/h}$ relative to the ground. To a passenger sitting inside the train, your velocity is $5 \text{ km/h}$. To a station agent standing on the platform, your velocity is $105 \text{ km/h}$ [3].
Taking the time derivative of our transformation equation and utilising our invariant clock ($dt' = dt$), we arrive at the classic velocity addition rule [3]:
$$\mathbf{v}' = \mathbf{v} - \mathbf{u}$$
Layer 3: Acceleration is Absolute(Intertial Frames)
If position and velocity are entirely relative to your choice of frame, is everything relative? Not quite. This brings us to the most powerful realisation in classical mechanics.
Acceleration is defined as the time rate of change of velocity: $\mathbf{a} = \frac{d\mathbf{v}}{dt}$ [3]. In an inertial frame, the relative frame velocity $\mathbf{u}$ is constant, meaning its time derivative is zero ($\frac{d\mathbf{u}}{dt} = 0$) [3].
When we differentiate our velocity equation, the frame velocity term vanishes [3]:
$$\mathbf{a}' = \frac{d\mathbf{v}'}{dt} = \frac{d\mathbf{v}}{dt} - 0 \implies \mathbf{a}' = \mathbf{a}$$
So, in all $\text{Inertial Frames}$ that are moving with constant velocity, acceleration is absolute.
The Sprinter, the Coach, and the Jet: Real-World Invariance
Acceleration is invariant across all inertial frames [3]! Whether you measure a sprinter exploding out of the starting blocks from a stationary track, a moving car, or a cruising aeroplane, everyone measures the exact same acceleration value.
This mathematical invariance is precisely why Isaac Newton anchored his Second Law ($\mathbf{F} = m\mathbf{a}$) to acceleration rather than velocity [3]. Because acceleration is absolute among inertial observers, the foundational laws of physics hold true everywhere.
Layer 4: When Acceleration Breaks Relative Motion (Non-Inertial Frames)
What happens when your observation frame itself accelerates? Imagine standing inside a subway car that abruptly slams on its brakes.
Slamming on the Brakes: Why Coffee Cups "Fly"
When frame $S'$ accelerates at $\mathbf{a}_{\text{frame}} \neq 0$ relative to the ground, our velocity derivative yields an accelerating kinematic relation [3]:
$$\mathbf{a}' = \mathbf{a} - \mathbf{a}_{\text{frame}}$$
If you are a passenger sitting inside that braking car, you are trapped inside a non-inertial frame. If you try to apply Newton's Second Law ($\mathbf{F} = m\mathbf{a}$) using only the visible forces around you, your equations will fail. Objects will appear to accelerate forward out of nowhere—like a coffee cup sliding off the dashboard—even though no physical hand pushed it. It all went well as long as the car was Inertial, going forward at a constant speed.
Bridging Kinematics to Dynamics: Introducing Fictitious Forces
To fix this broken equation from inside the car, we can take our kinematic acceleration relation and multiply both sides by mass ($m$) [4]:
$$m\mathbf{a}' = m\mathbf{a} - m\mathbf{a}_{\text{frame}}$$
Recognising that real physical forces generate true acceleration ($\mathbf{F} = m\mathbf{a}$), we can substitute that in and rearrange the terms [4]:
$$\mathbf{F}' = \mathbf{F} - m\mathbf{a}_{\text{frame}}$$
To make Newton's laws work locally inside your accelerating car without breaking reality, physicists shift the frame acceleration term to the other side, defining it as a fictitious force ($\mathbf{F}_{\text{fictitious}} = -m\mathbf{a}_{\text{frame}}$) [4]:
$$\mathbf{F}' + \mathbf{F}_{\text{fictitious}} = \mathbf{F}$$
These phantom forces—such as the inertial push you feel when a car brakes or the centrifugal pull on a merry-go-round—are not caused by physical interactions with other objects. Instead, they are mathematical correction terms required to compensate for the fact that your observation deck is accelerating [4].
Beyond Classical Limits: What Happens Near Light Speed?
Classical kinematics provides an airtight framework for everyday speeds, but Axiom 0 contains a hidden vulnerability. As physicist Albert Einstein demonstrated, time does not tick universally across the cosmos when speeds approach the speed of light ($c$) [5].
At near-light speeds, the Galilean transformation breaks down, requiring different equations (the Lorentz transformation), in which time dilates and length contracts. Newtonian absolute time was merely a low-speed approximation of a relativistic universe [5].
Motion is never an isolated property; it is an ongoing dialogue between an object and the observer watching it [1]. The next time you feel completely stationary, remember that you are participating in a vast, interconnected web of moving frames.
References & Suggested Reading
- Galilei, G. (1632). Dialogue Concerning the Two Chief World Systems. University of California Press.
- Resnick, R., Halliday, D., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
- Taylor, J. R. (2005). Classical Mechanics. University Science Books.
- Arons, A. B. (1997). Teaching Introductory Physics. John Wiley & Sons.
- Taylor, E. F., & Wheeler, J. A. (1992). Spacetime Physics: Introduction to Special Relativity (2nd ed.). W. H. Freeman.
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