Chapter 4 Lecture: Relative Motion- Part 1

Kinematic Reference Frames and Relative Velocity Concepts

Relative Nature of Velocity:
- Velocity is not an absolute quantity; it is inherently relative and depends strictly on the frame of reference or coordinate system from which an object is observed.
- Two distinct observers situated in different reference frames can measure two completely different velocities for the exact same physical object.
- The exact physical description of an object's spatial motion depends on the specific coordinate system defined by the observer.
Definition of Frame of Reference:
- A frame of reference is a defined coordinate system used to quantify and describe the position and motion of physical objects.
- Reference frames are typically attached or fixed to a specific observer, physical body, or location.
Earth / Ground Frame of Reference:
- Standard kinematic analysis routinely uses a coordinate system fixed to the physical surface of the Earth, commonly designated as the ground frame of reference.
- Motion observed relative to stationary surroundings (such as the ground or an airport floor) is evaluated within this ground frame.
Airport Walkway Thought Experiment:
- Physical Configuration:
- A moving walkway operates at a constant velocity of $1\,\text{m/s}$ in the positive $x$ -direction relative to the airport floor.
- A woman stands completely stationary relative to the surface of the moving walkway.
- A man walks forward along the moving walkway ahead of the standing woman at a constant speed of $1\,\text{m/s}$ relative to the walkway.
- A third observer sits stationary on a bench on the airport floor (ground frame), watching both individuals pass by.
- Coordinate System Assignments:
- Moving Frame ( $b$ Frame / Primed Coordinate System): A coordinate system designated as $x', y'$ with origin $O'$ attached to the moving walkway and centered on the stationary woman.
- Fixed Frame ( $a$ Frame / Non-Primed Coordinate System): A coordinate system designated as $x, y$ anchored to the ground observer seated on the airport floor.
- Measured Velocities across Reference Frames:
- From the standing woman's reference frame ( $b$ frame), the man is observed moving away at a constant velocity of $1\,\text{m/s}$ in the positive $x$ -direction.
- From the ground observer's reference frame ( $a$ frame), the walkway moves at $1\,\text{m/s}$ and the man moves at $1\,\text{m/s}$ relative to the walkway.
- The total velocity of the man observed from the ground frame is the vector sum of both speeds: $1\,\text{m/s} + 1\,\text{m/s} = 2\,\text{m/s}$ in the positive $x$ -direction.

Everyday Physical Examples and Reference Frame Dynamics

Bus Passenger Kinematics:
- An individual moving back and forth inside a moving transit bus exhibits different velocities depending on the observer's frame:
- An observer seated inside the bus measures the passenger's low walking velocity relative to the interior bus frame.
- An observer sitting outside on a roadside bus bench looking through the bus window measures the passenger moving at the combined velocity of the bus's road motion plus the internal walking motion.
Automotive Driving Scenarios:
- Stationary Vehicle Observer: A driver stopped at a red light/stop sign in an at-rest vehicle frame observes another car accelerate through a green light at $20\,\text{mph}$ . Because the observer frame is stationary relative to the ground, the observed vehicle speed matches the ground frame speed of $20\,\text{mph}$ .
- Highway Overtaking Dynamics:
- A car cruises on a freeway at $65\,\text{mph}$ (or $80\,\text{mph}$ ).
- A second vehicle passes the cruising car with a relative overtaking speed of $20\,\text{mph}$ (or $25\,\text{mph}$ ) measured from the driver's frame.
- Relative to the ground frame, the speed of the passing vehicle is the sum of the cruising vehicle speed and the relative passing speed: $65\,\text{mph} + 20\,\text{mph} = 85\,\text{mph}$ (or $65\,\text{mph} + 25\,\text{mph} = 90\,\text{mph}$ ).
- Reversing Vehicle Reference Frame:
- A driver passes an open parking spot at a school or mall, stops, and shifts into reverse, driving backward at $20\,\text{mph}$ relative to the ground.
- Relative to the reversing driver's frame of reference, a stationary car parked nearby appears to move forward at $20\,\text{mph}$ .
- Sensory Relative Motion Optical Illusion:
- Passengers or drivers resting in a stationary car or bus frequently experience a false illusion of rolling backward.
- This optical effect occurs when adjacent vehicles on both sides roll forward simultaneously at identical speeds.
- Visual tracking of adjacent moving reference frames tricks the sensory system into perceiving backward motion of the static vehicle relative to surrounding traffic.
Validity of Earth Frame Approximations:
- Physical Reality: The Earth is not a static, fixed system; it undergoes axial rotation and orbital movement through space.
- Planetary Scale Effects: For advanced engineering, atmospheric systems, or trajectories spanning planetary scale lengths comparable to the Earth's radius $R_E$ , the rotation of the Earth introduces non-inertial relative motion effects.
- Localized Kinematic Approximations: For standard physical objects moving over terrestrial distances much smaller than the dimensions of the Earth, the Earth's rotation can be safely ignored, allowing the ground to be treated as a fixed coordinate system.

Mathematical Derivation of Galilean Relative Velocity Transformations

Governing Assumptions and Boundary Conditions:
- Galilean relative motion equations strictly apply under the following specific physical conditions:
1. Common Aligned Axis: Motion occurs along a shared single coordinate direction (e.g., motion aligned along the $x$ -axis).
2. Uniform Relative Velocities: Reference frames move relative to one another at constant velocity (zero relative acceleration). Accelerated reference frames are governed by non-inertial Einsteinian relativity.
3. Non-Relativistic Speeds: Object speeds $v$ are vastly smaller than the speed of light $c$ ( $v \ll c$ ).
4. Standard Kinematic Assumptions: The object is approximated as a point particle, atmospheric air resistance is neglected, Earth's curvature is ignored, and temporal duration $t$ is absolute and identical across all reference frames (no relativistic time dilation).
Position Vector Derivation:
- Let $p$ represent an observed point particle or physical object.
- Let $a$ denote Reference Frame $A$ with origin $O_a$ .
- Let $b$ denote Reference Frame $B$ with origin $O_b$ and coordinate axes $x_b, y_b, z_b$ .
- Spatial Vector Definitions:
- $r_{p/b}$ : Position vector locating object $p$ relative to origin $O_b$ of Frame $b$ .
- $r_{p/a}$ : Position vector locating object $p$ relative to origin $O_a$ of Frame $a$ .
- $r_{b/a}$ : Position vector locating the origin $O_b$ of Frame $b$ relative to origin $O_a$ of Frame $a$ .
- Vector Addition and Parallelogram Method:
- Connecting the displacement vectors tip-to-tail establishes the resultant vector equation: $r_{p/a} = r_{b/a} + r_{p/b}$
- By applying the commutative property of vector addition ( $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ ): $r_{p/a} = r_{p/b} + r_{b/a}$
Time Derivative and Velocity Transformation:
- Instantaneous velocity is defined as the time derivative of spatial position: $v = \frac{dr}{dt}$ .
- Differentiating the position vector equation with respect to invariant time $t$ : $\frac{d}{dt}(r_{p/a}) = \frac{d}{dt}(r_{p/b} + r_{b/a})$
- Applying the sum rule for differentiation: $\frac{d}{dt}(r_{p/a}) = \frac{d}{dt}(r_{p/b}) + \frac{d}{dt}(r_{b/a})$
- Evaluating these derivatives yields Galileo's Relative Velocity Equation: $v_{p/a} = v_{p/b} + v_{b/a}$

Subscript Formalism and Application Procedures

Subscript Formalism Rules:
- Notation Definition:
- $v_{p/a}$ : Velocity of object $p$ with respect to Frame $a$ .
- $v_{p/b}$ : Velocity of object $p$ with respect to Frame $b$ .
- $v_{b/a}$ : Velocity of Frame $b$ with respect to Frame a$.\n * Subscript Cancellation Mechanics:\n * In the standard additive formulation v_{p/a} = v_{p/b} + v_{b/a} $, the inner subscripts ($ b $and$ b) are adjacent.\n * These adjacent inner subscripts mathematically cancel out.\n * The remaining outer subscripts (p $and$ a) dictate the index ordering of the resultant velocity vector on the left side of the equation.\n * This strict notation prevents algebraic errors when determining whether to add or subtract reference frame velocities.\n\n* **Step-by-Step Problem Solving Protocol**:\n 1. Define the physical system entities: Identify observed particle p $, moving reference frame$ b $, and fixed reference frame$ a\n 2. Write down Galileo's relative velocity formalism equation directly: v_{p/a} = v_{p/b} + v_{b/a}\n 3. Extract component velocities along the shared axis with proper sign conventions relative to defined origins.\n 4. Substitute component scalar values into the equation to compute the unknown target relative velocity.\n\n* **Worked Example (Airport Walkway)**:\n * System Identification: Particle p = m $(man), Frame$ b = \text{walkway frame} $, Frame$ a = \text{ground frame}.\n * Galilean Formalism Setup: v_{m/a} = v_{m/b} + v_{b/a}\n * Parameter Extraction:\n * Velocity of man relative to walkway: v_{m/b} = +1\,\text{m/s}\n * Velocity of walkway relative to ground: v_{b/a} = +1\,\text{m/s}\n * Algebraic Computation:\n v_{m/a} = 1\,\text{m/s} + 1\,\text{m/s} = 2\,\text{m/s}\n * Conclusion: The ground observer measures the man traveling at 2\,\text{m/s} $in the positive$ x$$-direction.