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Mechanics
Galilean Transformation Equations
Galilean transformations are a set of equations used in classical mechanics to relate the space and time coordinates of an event as observed by two different inertial reference frames moving at a constant relative velocity. They form the mathematical foundation of Newtonian mechanics and uphold the principle of absolute time.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Understanding position, velocity, and acceleration transformations between inertial frames, invariance of Newton's laws under Galilean transformations, and concept of absolute time.
PGT extension
Transformation of non-inertial frames, introduction of fictitious forces (centrifugal and Coriolis forces), and the limitations of Galilean relativity in electrodynamics leading to Lorentz transformations.
Prerequisites
Inertial and non-inertial reference frames
Vector addition and relative velocity in kinematics
Newton's laws of motion
Postulates of Galilean Relativity
Space is isotropic and homogeneous, and time is absolute, meaning the time interval between two events is identical for all observers regardless of their relative state of motion.
Newtonian mechanics and the laws of motion are form-invariant (covariant) under Galilean transformations.
An inertial reference frame is one in which Newton's first law of motion holds true, moving at zero acceleration.
Space and Time Coordinates
Let frame S' move with a constant velocity v along the positive x-axis relative to frame S.
Assuming synchronous clocks such that t = t' = 0 when the origins coincide, the spatial coordinates are related by x' = x - vt, y' = y, and z' = z.
Time is universal, so the temporal transformation is simply t' = t.
Velocity Transformation
By differentiating the position transformation equations with respect to time, the velocity addition theorem in classical mechanics is obtained as u' = u - v.
Acceleration remains invariant under Galilean transformation because the relative velocity v is constant, yielding a' = a.
Forces derived from relative positions (such as gravitational or electrostatic forces) are invariant under Galilean transformations.
Limitations in Electrodynamics
Galilean transformations predict that the speed of light is frame-dependent, contrary to the principle of invariance of the speed of light.
Maxwell's equations of electromagnetism are not invariant under Galilean transformations, failing the classical relativity principle for electromagnetic phenomena.
At velocities approaching the speed of light, Galilean mechanics breaks down and must be replaced by Einstein's special theory of relativity using Lorentz transformations.
Derivations
Derivation of Position and Velocity Transformation Equations
Conditions: Frame S' moves with constant velocity v along the x-axis relative to inertial frame S.
Let an event P have coordinates (x, y, z, t) in frame S and (x', y', z', t') in frame S'.
At time t = 0, origins of both frames coincide. After time t, the origin of S' has moved by a distance vt along the x-axis.
From vector geometry, the x-coordinate in S is $x = x' + vt$, which rearranges to x' = x - vt. The y and z coordinates remain unaffected: y' = y, z' = z.
Assuming absolute time, t' = t.
Differentiating position coordinates with respect to time yields velocity components: u'_$x = u_x - v$, u'_$y = u_y$, u'_$z = u_z$.
Result: x' = x - vt, y' = y, z' = z, t' = t; u' = u - v
Transformation of Acceleration
Conditions: Frame S' moves with constant linear velocity v relative to frame S.
Start with the velocity transformation equation: u'_$x = u_x - v$.
Differentiate the velocity equation with respect to time t (noting that dt' = dt): a'_$x = d(u'_x)/dt' = d(u_x - v)/dt$.
Since v is a constant relative velocity, its time derivative is zero (dv/dt = 0).
Thus, a'_$x = du_x/dt = a_x$.
Similarly, a'_$y = a_y$ and a'_$z = a_z$, proving acceleration is invariant.
Result: a' = a (Acceleration is invariant under Galilean transformations)
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Position Transformation (X-axis)
$$x' = x - vt$$
x, x' = positions in S and S'; v = relative velocity; t = time m
Constant relative velocity along x-axis
Position Transformation (Y and Z axes)
$$y' = y, \quad z' = z$$
y, y', z, z' = transverse coordinates m
Motion restricted to x-axis
Time Transformation
$$t' = t$$
t, t' = time coordinates in respective frames s
Universal absolute time assumption
Velocity Addition Theorem
$$u' = u - v$$
u, u' = velocities in S and S'; v = frame velocity m/s
Low velocities (u << c)
Acceleration Invariance
$$a' = a$$
a, a' = accelerations in S and S' m/s^2
Inertial frames with constant relative velocity
Inverse Position Transformation
$$x = x' + vt$$
x, x' = positions; v = relative velocity; t = time m
Transforming from S' back to S
Inverse Velocity Transformation
$$u = u' + v$$
u, u' = velocities; v = relative velocity m/s
Transforming from S' back to S
Force Invariance in Newtonian Mechanics
$$F' = F$$
F, F' = forces measured in frames S and S' N
Mass is invariant and acceleration is invariant
Graphs and experiments
Space-Time (X-t) Diagram for Galilean Transformation
Draw: A Cartesian plot with position x on the vertical axis and time t on the horizontal axis, showing worldlines of a stationary particle (vertical line) and a moving particle (sloped line) for both S and S' frames.
Interpret: Illustrates how spatial coordinates shift linearly with time between relative moving frames while maintaining a single, universal time axis.
Exam use: Tested conceptually in TGT/PGT to distinguish Galilean space-time geometry from Minkowski space-time diagrams.
Michelson-Morley Experiment (Null Result)
Draw: Schematic diagram of an optical interferometer with orthogonal arms used to detect the Earth's motion through the hypothetical luminiferous ether.
Interpret: Demonstrated the absence of an ether drift, proving that Galilean velocity addition fails for the speed of light.
Exam use: Crucial historical background question explaining why Galilean transformations were superseded by Lorentz transformations.
Relative Velocity Vector Plot
Draw: A vector triangle showing particle velocity u, frame velocity v, and relative velocity u' = u - v.
Interpret: Represents the linear vector superposition principle of classical kinematics.
Exam use: Used in numerical problem-solving for river-boat and rain-umbrella problems.
Solved numericals
TGT core
A train is moving at a uniform velocity of 60 km/h due east relative to the ground. A passenger walks inside the train at 5 km/h due west relative to the train. Find the velocity of the passenger relative to the ground.
Identify the given quantities: Velocity of train (frame S' relative to S) $v = +60 km/h (east)$. Velocity of passenger relative to train u' = -5 km/h (west).
Substitute the values with proper sign convention (taking east as positive): $u = -5 + 60 = +55 km/h$.
Answer: 55 km/h due east
PGT extension
Two reference frames S and S' have a relative velocity v along the x-axis. A particle moves with velocity components $u_x = 3 m/s$ and $u_y = 4 m/s in frame S$. If the transformed velocity component in frame S' along the x-axis is u'_$x = 1 m/s$, determine the relative velocity v of the frames and the transformed y-component of velocity u'_y.
Use the velocity transformation relation: u'_$x = u_x - v$.
Substitute the known values: $1 = 3 - v$, which yields $v = 2 m/s$.
Use the transverse velocity transformation relation: u'_$y = u_y$.
In an inertial frame S, a body of mass 2 kg experiences a constant acceleration $a = 3 i-hat + 4 j-hat m/s^2$. If observed from a frame S' moving with a uniform velocity of 10 i-hat m/s relative to S, what is the acceleration of the body and the net force experienced by the body in frame S'?
Use the acceleration invariance property under Galilean transformations: a' = a.
Therefore, acceleration in frame S' is a' = 3 i-hat + 4 j-hat m/s^2.
Apply Newton's second law in frame S': F' = m * a'.
Multiply mass m = 2 kg by vector a': F' = 2 * (3 i-hat + 4 j-hat) = 6 i-hat + 8 j-hat N.
Answer: Acceleration a' = 3 i-hat + 4 j-hat m/s^2, Force F' = 6 i-hat + 8 j-hat N
Assumptions and limitations
Assumption: Time is absolute and flows uniformly for all observers regardless of relative motion (t' = t).
Limitation: Fails at high speeds comparable to the speed of light (u ~ c), where relativistic effects like time dilation and length contraction become significant.
Limitation: Invalid in non-inertial accelerating frames unless fictitious forces are explicitly introduced.
Common misconceptions
Common error: Time changes between different inertial frames in Galilean relativity.
Correct: Time is strictly absolute (t' = t) in Galilean transformations.
Classical mechanics assumes a universal, independent time scale for all observers, unlike Einstein's special relativity.
Common error: Acceleration changes when observed from a uniformly moving reference frame.
Correct: Acceleration is completely invariant under Galilean transformations.
Since the relative velocity between inertial frames is constant, its time derivative is zero, leaving acceleration unchanged (a' = a).
Common error: Maxwell's equations are invariant under Galilean transformations.
Correct: Maxwell's equations are not invariant under Galilean transformations.
Electrodynamics requires Lorentz transformations because the speed of light is constant in all inertial frames.
Quick revision
Galilean transformations relate space-time coordinates between inertial frames via x' = x - vt and t' = t.
Position Transformation (X-axis): x' = x - vt
Position Transformation (Y and Z axes): y' = y, \quad z' = z
Must remember
Time is absolute in Galilean relativity, meaning $t' = t$ for all inertial observers.
The spatial position transformation along the x-axis is given by $x' = x - vt$.
Acceleration remains invariant under Galilean transformations, such that $a' = a$.
Maxwell's equations of electromagnetism are not invariant under Galilean transformations.
The velocity addition theorem in classical mechanics is expressed as $u' = u - v$.
Glossary
Galilean TransformationA set of equations used to relate the space and time coordinates of two inertial frames moving at a constant relative velocity.
Absolute TimeThe assumption that the time interval between events is identical for all observers regardless of their relative motion ($t' = t$).
Inertial Reference FrameA reference frame in which Newton's first law of motion holds true and which moves at zero acceleration.
Velocity Addition TheoremThe classical law relating velocities measured in different inertial frames, given by $u' = u - v$.
Galilean vs. Lorentz Transformations
Parameter
Galilean Transformation
Lorentz Transformation
Time Nature
Absolute ($t' = t$)
Relative and dependent on velocity
Speed of Light
Frame-dependent ($c' \neq c$)
Invariant ($c' = c$ in all frames)
Applicability
Low velocities ($u \lt\lt c$)
All velocities up to the speed of light
Practice Quiz
Choose an option to reveal the explanation.
1. What is the temporal transformation relation between two inertial frames in Galilean relativity?
Explanation: In Galilean relativity, time is assumed to be absolute and universal, meaning $t' = t$ for all observers.
2. If frame $S'$ moves with a constant velocity $v$ along the positive x-axis relative to frame $S$, what is the spatial coordinate transformation for $x'$?
Explanation: The Galilean spatial coordinate transformation along the direction of relative motion is given by $x' = x - vt$.
3. How do the transverse coordinates $y$ and $z$ transform under Galilean relativity when motion is confined to the x-axis?
Explanation: Since motion is restricted exclusively along the x-axis, the transverse dimensions perpendicular to the motion remain unaffected, so $y' = y$ and $z' = z$.
4. What is the classical velocity addition theorem derived from Galilean transformations?
Explanation: Differentiating the position transformation $x' = x - vt$ with respect to time yields the velocity addition relation $u' = u - v$.
5. What happens to acceleration when measured from two different inertial frames related by a Galilean transformation?
Explanation: Since the relative velocity $v$ between inertial frames is constant, its time derivative is zero, leaving acceleration completely invariant ($a' = a$).
6. Which of the following equations is NOT invariant under Galilean transformations?
Explanation: Maxwell's equations of electrodynamics are not invariant under Galilean transformations, which led to the development of Lorentz transformations.
7. What is the inverse position transformation for $x$ in terms of $x'$ and $t$?
Explanation: Transforming back from frame $S'$ to $S$ yields the inverse position relation $x = x' + vt$.
8. What is the inverse velocity transformation relating $u$ to $u'$?
Explanation: The inverse velocity transformation is obtained from the inverse position relation, giving $u = u' + v$.
9. Why do Galilean transformations fail at speeds approaching the speed of light ($c$)?
Explanation: Galilean transformations lead to classical velocity addition where the speed of light would be frame-dependent, contradicting electromagnetic theory and experiments.
10. What defines an inertial reference frame?
Explanation: An inertial reference frame is defined as one in which Newton's first law of motion holds true and which moves at zero acceleration.
11. In PGT extensions, when transforming into a non-inertial reference frame accelerating at $A$, what type of forces must be introduced?
Explanation: To apply Newton's laws in non-inertial frames, fictitious forces (such as $-mA$) must be introduced to account for the frame's acceleration.
12. Which force components arise when transforming into a rotating reference frame?
Explanation: Rotating frames introduce non-inertial terms into the equation of motion, specifically the centrifugal and Coriolis forces.
13. If mass is assumed to be invariant under Galilean transformations ($m' = m$), how does force $F'$ transform compared to $F$?
Explanation: Since mass and acceleration are both invariant under Galilean transformations ($m' = m$ and $a' = a$), force derived from relative positions is also invariant ($F' = F$).
14. What does the assumption of space being isotropic and homogeneous imply in Galilean relativity?
Explanation: Isotropy means space looks the same in all directions, and homogeneity means it is uniform at all points.
15. Which framework replaces Galilean mechanics at high velocities near the speed of light?
Explanation: At velocities approaching the speed of light, Galilean mechanics breaks down and is replaced by Einstein's special theory of relativity.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. Under a Galilean transformation, if frame $S'$ moves at velocity $v$ along the x-axis, the time interval $\Delta t = t_2 - t_1$ between two events is:
Explanation: Because time is absolute in Galilean relativity ($t' = t$), the time interval between any two events is identical in all inertial frames, so $\Delta t' = \Delta t$.
2. If an object moves with velocity $u$ in frame $S$, and frame $S'$ moves with velocity $v$ in the same direction, the velocity $u'$ measured in $S'$ according to Galilean transformation is:
Explanation: The classical velocity addition theorem states that $u' = u - v$ when transforming from frame $S$ to $S'$.
3. Which of the following physical quantities is invariant under Galilean transformations?
Explanation: Position and velocity depend on the relative frame velocity and change under Galilean transformations, but acceleration ($a' = a$) and force remain invariant.
4. Why are Maxwell's equations incompatible with Galilean transformations?
Explanation: Maxwell's equations yield a constant wave speed ($c$) for electromagnetic waves, which fails to transform correctly under Galilean equations and requires Lorentz transformations.
5. In PGT mechanics, when writing Newton's second law in a uniformly accelerating reference frame with acceleration $A$, the effective force is modified by adding:
Explanation: In a linearly accelerating reference frame, an additional inertial or fictitious force of magnitude $-mA$ must be introduced.
6. Which of the following conditions must be satisfied for two reference frames to be considered strictly inertial?
Explanation: Inertial frames are unaccelerated relative to each other, moving with constant velocity, such that Newton's laws hold in all such frames.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary postulate regarding time in classical Newtonian mechanics and Galilean relativity?
Explanation: Classical mechanics assumes absolute time, where $t' = t$ for any observer regardless of relative motion.
2. If a particle's position in frame $S$ is given by $x = 5t^2$, what is its position $x'$ in frame $S'$ moving at velocity $v$ along the x-axis?
Explanation: Using $x' = x - vt$ and substituting $x = 5t^2$ with $t' = t$, we get $x' = 5t^2 - vt$.
3. If the velocity of a body in frame $S'$ is $u' = 10\text{ m/s}$ and the frame moves at $v = 5\text{ m/s}$ in the positive x-direction, what is its velocity $u$ in frame $S$?
Explanation: Using the inverse velocity transformation $u = u' + v$, we find $u = 10 + 5 = 15\text{ m/s}$.
4. Which of the following statements about Newtonian mechanics under Galilean transformations is correct?
Explanation: Newtonian mechanics and laws of motion are form-invariant under Galilean transformations.
5. What is the transformation equation for the z-coordinate when frame $S'$ moves relative to $S$ along the x-axis?
Explanation: Since relative motion is strictly along the x-axis, the z-coordinate remains unchanged ($z' = z$).
6. Why does Galilean relativity break down at very high velocities?
Explanation: Galilean velocity addition implies light speed would vary with observer velocity, which contradicts experimental results establishing the invariance of $c$.
7. In PGT mechanics, what is the mathematical expression for the Coriolis force in a rotating coordinate system with angular velocity $\vec{\omega}$ and particle velocity $\vec{v}'$?
Explanation: The Coriolis force in a rotating non-inertial frame is given by $-2m(\vec{\omega} \times \vec{v}')$, while the centrifugal force is $-m\vec{\omega} \times (\vec{\omega} \times \vec{r})$.
8. If an observer in frame $S$ measures an acceleration of $4\text{ m/s}^2$ for a moving particle, what acceleration does an observer in frame $S'$ (moving at constant velocity $v$) measure?
Explanation: Acceleration is invariant under Galilean transformations between inertial frames, so $a' = a = 4\text{ m/s}^2$.
9. Which transformation set replaces Galilean transformations when dealing with electromagnetic phenomena at high speeds?
Explanation: Lorentz transformations preserve Maxwell's equations and the invariance of the speed of light, replacing Galilean transformations at high velocities.
10. What does the inverse position transformation $x = x' + vt$ signify physically?
Explanation: The inverse position transformation allows an observer in frame $S'$ to express coordinates back in terms of frame $S$.