Relative motion Knowledge-Based Study

Study notes

1. Frames of Reference

   Motion is not absolute; it is always described relative to a chosen point of view called a frame of reference. A frame of reference consists of a coordinate system and a clock used to measure positions and times of events.

   For example, consider a passenger sitting on a train moving smoothly forward at 20 m/s:

   • Relative to another seated passenger on the train, the passenger is at rest.
   • Relative to an observer standing on the ground outside, the passenger moves forward at 20 m/s.

   Both observers are correct within their own reference frames. Consequently, whenever we state a displacement, velocity, or acceleration, an underlying reference frame is either explicitly specified or implicitly assumed (often the ground or Earth).

   One-Dimensional Relative Motion

   When two objects move along the same straight line, determining their relative velocity involves simple algebraic subtraction along that coordinate axis.

   Let $v_A$ and $v_B$ represent the velocities of objects A and B with respect to a common reference frame, such as the ground. The velocity of object A relative to object B, denoted as $v_{A/B}$, is defined as:

   $$v_{A/B} = v_A - v_B$$

   Here, sign conventions indicate directions (for instance, choosing right as positive and left as negative):

   • Objects moving in the same direction: If car A moves east at 25 m/s and car B moves east at 15 m/s, the velocity of A relative to B is $v_{A/B} = 25\text{ m/s} - 15\text{ m/s} = +10\text{ m/s}$ (east). Car B perceives car A pulling away at 10 m/s.
   • Objects moving in opposite directions: If car A moves east at 25 m/s and car C moves west at 15 m/s (so $v_C = -15\text{ m/s}$), then $v_{A/C} = 25\text{ m/s} - (-15\text{ m/s}) = +40\text{ m/s}$. Car C perceives car A approaching at 40 m/s.

   Vector Addition and Subtraction

   When motion occurs in two dimensions, such as a boat crossing a flowing river or an aircraft flying in crosswinds, velocities must be treated as full vector quantities.

   If $\vec{v}_{A/G}$ is the velocity of object A relative to the ground and $\vec{v}_{B/G}$ is the velocity of object B relative to the ground, the velocity of A relative to B is:

   $$\vec{v}_{A/B} = \vec{v}_{A/G} - \vec{v}_{B/G}$$

   Equivalently, vector addition allows combining relative velocities sequentially across intermediate frames:

   $$\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$$

   Example: Boat Crossing a River

   Consider a boat moving with a velocity $\vec{v}_{B/W}$ relative to the water, while the river water flows with velocity $\vec{v}_{W/G}$ relative to the ground. The resultant velocity of the boat relative to the ground $\vec{v}_{B/G}$ is given by:

   $$\vec{v}_{B/G} = \vec{v}_{B/W} + \vec{v}_{W/G}$$

   If the boat heads perpendicular to the current (east) at 4.0 m/s and the river flows south at 3.0 m/s, the magnitude of the boat's velocity relative to the ground is:

   $$|\vec{v}_{B/G}| = \sqrt{(4.0\text{ m/s})^2 + (3.0\text{ m/s})^2} = 5.0\text{ m/s}$$

   The boat drifts downstream at an angle determined by the vector sum of both velocities.

   Summary of Core Principles

   • Observer Dependency: Position and velocity depend entirely on the chosen frame of reference.
   • Subscript Rule: For any three reference points or objects A, B, and C, the relation $\vec{v}_{A/C} = \vec{v}_{A/B} + \vec{v}_{B/C}$ holds consistently, where inner subscripts match.
   • Reciprocity: Swapping the observer reverses the sign of the relative velocity vector: $\vec{v}_{B/A} = -\vec{v}_{A/B}$.
   • Orthogonal Components: In two-dimensional problems, resolve each vector into horizontal ($x$) and vertical ($y$) components, apply the relative velocity equations independently to each axis, and recombine using the Pythagorean theorem and trigonometry.