Reference Frames: Inertial vs. Non-Inertial
In physics, reference frames are essential for describing the position and motion of objects.
Reference frame
A reference frame is a coordinate system that allows us to measure the position, velocity, and time of events.
Inertial Reference Frames
Inertial reference frame
An inertial reference frame is one where Newton’s first law holds true: an object at rest stays at rest, and an object in motion continues in a straight line at constant speed unless acted upon by a force.
Inertial frames are either at rest or moving with constant velocity.
Non-Inertial Reference Frames
Non-inertial reference frame
A non-inertial reference frame is one that is accelerating.
In these frames, objects appear to experience fictitious forces, such as the sensation of being pushed back in a car that accelerates forward.
Non-inertial frames require corrections for these fictitious forces to accurately describe motion.
Identifying Reference Frames
Consider these scenarios:
- A train moving at constant speed on a straight track is an inertial frame.
- A car accelerating around a curve is a non-inertial frame.
Which of the following are inertial frames?
- A satellite orbiting Earth
- A car moving at constant speed on a straight road
- A spinning merry-go-round
Galilean Transformations: Relating Two Reference Frames
- Galilean transformations provide a mathematical framework to relate the coordinates of an event in one inertial frame to another.
- These transformations assume that time is absolute and the same for all observers.
The Basic Equations
Consider two reference frames, $S$ and $S′$:
- $S$ is stationary.
- $S′$ moves with a constant velocity $v$ relative to $S$ along the x-axis.
The Galilean transformations are:
- Position transformation: $$x'=x-vt$$
- Time transformation: $$t'=t$$
- Imagine a train moving at $10 \text{ m s}^{-1}$.
- A passenger drops a ball, and it hits the floor at $x' = 2 \text{ m}$ and $t' = 1 \text{ s}$ in the train’s frame ($S'$).
- For an observer on the ground ($S$), the ball’s position is: $$x=x'+vt=2\ \mathrm{m}+10\ \mathrm{m/s}×1\ \mathrm{s}=12\ \mathrm{m}$$
- The time remains the same: $$t=t'=1\ \mathrm{s}$$
Velocity Addition: Understanding Relative Velocity
Galilean transformations also help us understand how velocities are perceived differently in different frames.
The Galilean Velocity Addition Formula
If an object moves with velocity $u'$ in frame $S'$, its velocity $u$ in frame $S$ is given by:
$$u=u'+v$$
- A train moves at $20 \text{ m s}^{-1}$ relative to the ground.
- A passenger throws a ball forward at $5 \text{ m s}^{-1}$ relative to the train.
- Velocity of the ball relative to the ground: $$u=u'+v=5 \text{ m s}^{-1}+20\ \mathrm{m/s}=25 \text{ m s}^{-1}$$
A boat moves at $4 \text{ m s}^{-1}$ relative to a river flowing at $2 \text{ m s}^{-1}$. What is the boat’s velocity relative to the riverbank?
Limitations of Galilean Relativity
Galilean relativity works well for everyday speeds but breaks down at velocities approaching the speed of light.
Why Galilean Relativity Fails at High Speeds
- Galilean transformations assume that time is absolute and velocities add linearly.
- However, experiments show that the speed of light is constant for all observers, regardless of their motion.
This led to the development of Einstein’s theory of special relativity, which replaces Galilean transformations with Lorentz transformations.
- Imagine a spaceship moving at 0.9c (90% the speed of light) relative to Earth.
- It emits a light beam forward.
- According to Galilean relativity, the light’s speed would be 1.9c for an Earth observer.
- However, experiments confirm that the light’s speed remains c, not 1.9c.
