Relative Motion and Galilean Transformation

In the study of classical mechanics, motion is never an isolated property of an object; rather, it is a relationship bet

In the study of classical mechanics, motion is never an isolated property of an object; rather, it is a relationship between an object and its environment. To describe a physical process accurately, one must first establish a perspective. What appears to be stationary to one observer may appear to be in rapid motion to another. To quantify these differing observations, physics relies on a mathematical framework known as relative motion, with the Galilean Transformation serving as the fundamental bridge between different viewpoints.
To measure position, velocity, and acceleration, we require a coordinate system, which in physics is termed a Reference Frame. Because there is no "absolute center" or "stationary point" in the universe, every measurement of motion is inherently relative to a chosen frame.

  • The Relativity of Motion: Consider a passenger sitting in a high-speed train moving at a constant velocity. To the passenger, the seat beside them is stationary. However, to a pedestrian standing on the station platform, that same passenger is moving at a high velocity. Both observations are correct within their respective frames.
  • Inertial Reference Frames: Not all frames are created equal. An Inertial Reference Frame is a specific type of frame in which Newton’s First Law (the Law of Inertia) holds true. In such a frame, an object will remain at rest or continue to move in a straight line at a constant speed unless acted upon by an external force.
  • Relative Motion: This refers to the kinematic relationship between the motion of an object as observed from two or more different reference frames. The goal of classical kinematics is to derive the mathematical rules that allow us to translate observations from one frame to another.

Mathematical Derivation of the Galilean Transformation

To link two inertial reference frames that are moving at a constant velocity relative to one another, we use the Galilean Transformation.

Let us assume two inertial frames, $S$ and $S'$:

  • Frame $S$ is considered the "stationary" frame with coordinates $(x, y, z, t)$.
  • Frame $S'$ is moving with a constant velocity $v$ along the positive $x$-axis of frame $S$, with coordinates $(x', y', z', t')$.
  • At the initial time $t = t' = 0$, the origins of both frames ($O$ and $O'$) coincide.

In the classical worldview, space and time are treated as absolute. This means that time flows at the same rate for all observers regardless of their motion ($t = t'$). Based on this assumption, we can derive the spatial transformation equations:

  1. Position Transformation:
    Since the origin $O'$ of frame $S'$ is located at $x = vt$ within frame $S$, the position of any point in the moving frame is shifted by that distance. Thus:
    $$x' = x - vt$$
    Conversely, to find the position in the stationary frame:
    $$x = x' + vt$$
    Because there is no relative motion along the $y$ and $z$ axes, the coordinates remain identical:
    $$y' = y$$
    $$z' = z$$

  2. Time Transformation:
    Under the principle of absolute time in classical mechanics:
    $$t' = t$$

Velocity Transformation and the Invariance of Acceleration

By applying calculus to the Galilean transformation equations, we can determine how velocities differ between observers.

Let the velocity components of a particle in frame $S$ be $u_x = \frac{dx}{dt}$, $u_y = \frac{dy}{dt}$, and $u_z = \frac{dz}{dt}$. The velocity components in frame $S'$ are $u_x'$, $u_y'$, and $u_z'$. By differentiating the position equations with respect to time, we get:

$$u_x' = \frac{dx'}{dt'} = \frac{d(x - vt)}{dt} = \frac{dx}{dt} - v = u_x - v$$
$$u_y' = \frac{dy'}{dt'} = \frac{dy}{dt} = u_y$$
$$u_z' = \frac{dz'}{dt'} = \frac{dz}{dt} = u_z$$

This result is known as the Galilean Velocity Addition Principle. It dictates that the velocity of an object in the stationary frame is the vector sum of its velocity in the moving frame and the velocity of the moving frame itself.

A more profound consequence arises when we examine acceleration. By differentiating the velocity components once more:

$$a_x' = \frac{du_x'}{dt'} = \frac{d(u_x - v)}{dt} = \frac{du_x}{dt} - 0 = a_x$$

Since $v$ is constant, its derivative is zero. Therefore, $a_x' = a_x$, $a_y' = a_y$, and $a_z' = a_z$. This demonstrates that acceleration is invariant under Galilean transformations. Regardless of whether you are in frame $S$ or $S'$, the acceleration of the object remains the same. This invariance is critical because it ensures that Newton’s Second Law ($\vec{F} = m\vec{a}$) maintains the same mathematical form in all inertial frames, upholding the principle of Galilean relativity.

Practical Application and Limitations

To visualize this, consider a common scenario:

Example: An aircraft carrier is cruising East at a velocity of $v = 10 \text{ m/s}$. A sailor on the deck is running East at a velocity of $u' = 5 \text{ m/s}$ relative to the ship. What is the velocity of the sailor as measured by an observer standing on the shore?

Solution:

  • Let the shore be the stationary frame $S$.
  • Let the aircraft carrier be the moving frame $S'$, where $v = 10 \text{ m/s}$.
  • The sailor's velocity in $S'$ is $u' = 5 \text{ m/s}$.
  • Using the velocity transformation $u = u' + v$:
    $$u = 5 \text{ m/s} + 10 \text{ m/s} = 15 \text{ m/s}$$
    The observer on the shore sees the sailor moving at $15 \text{ m/s}$.

The Boundaries of Classical Mechanics

While the Galilean transformation is an incredibly powerful tool for macroscopic engineering and everyday physics, it is not universal. Its validity rests on the assumption of absolute time and absolute space.

As velocities approach the speed of light ($c$), these assumptions fail. Experimental evidence shows that the Galilean velocity addition law does not hold at relativistic speeds. In such regimes, classical mechanics must be replaced by Einstein's Special Theory of Relativity, where the Lorentz Transformation takes the place of the Galilean Transformation to account for the constancy of the speed of light and the relativity of simultaneity. Nevertheless, for the vast majority of human endeavors, the Galilean framework remains the essential mathematical foundation for understanding the world in motion.