Introduction
Velocity is frame-dependent and must be measured relative to a reference frame. The Galilean transformation relates velocities in different inertial frames moving at constant relative velocity. Understanding relative motion is essential for problems involving multiple moving objects or observers, such as navigation, collision analysis, and reference frame transformations.
Galilean Velocity Transformation
If frame S' moves at constant velocity V relative to frame S, then an object's velocity measured in S (v) and in S' (v') are related by: v = v' + V or v' = v - V. This is classical velocity addition. For example, if a train moves at 30 m/s and a person walks at 2 m/s toward the front relative to the train, their ground speed is 32 m/s. All velocities are vectors and must be added/subtracted as vectors.
\nIntroduction
Velocity is frame-dependent and must be measured relative to a reference frame. The Galilean transformation relates velocities in different inertial frames moving at constant relative velocity. Understanding relative motion is essential for problems involving multiple moving objects or observers, such as navigation, collision analysis, and reference frame transformations.
Galilean Velocity Transformation
If frame S' moves at constant velocity V relative to frame S, then an object's velocity measured in S (v) and in S' (v') are related by: v = v' + V or v' = v - V. This is classical velocity addition. For example, if a train moves at 30 m/s and a person walks at 2 m/s toward the front relative to the train, their ground speed is 32 m/s. All velocities are vectors and must be added/subtracted as vectors.
\nRelative Velocity in Two Dimensions
In 2D problems, resolve all velocities into components. The velocity of A relative to B is v_AB = v_A - v_B where v_A and v_B are velocities relative to ground (or any common frame). To find the velocity of B relative to A: v_BA = -v_AB (opposite direction). Aircraft navigation uses this: ground velocity = air velocity + wind velocity. Boat crossing river: velocity relative to shore = velocity relative to water + velocity of water.
Crossing River Problem
Classic application: Boat with speed v in still water crosses river with current speed u. If boat heads at angle θ upstream: across-river component is vsinθ, downstream component is u - vcosθ. To cross directly (no downstream drift): point upstream such that vcosθ = u. Time to cross: t = w/(vsinθ) where w is river width. Minimum crossing time: head straight across (θ = 90°), though this results in downstream drift.
\nIntercept and Collision Problems
To determine if two objects collide or intercept: write position vectors as functions of time for both objects. Set r1(t) = r2(t) and solve for t. If solution exists with t > 0, collision occurs. For projectile interception (like a missile hitting a target), aim not at current position but at predicted future position. This requires solving simultaneous equations of motion for both objects.
Frame-Dependent and Frame-Independent Quantities
Position and velocity are frame-dependent (relative). Acceleration is frame-independent in inertial frames (all inertial observers measure same acceleration). Time and mass are absolute in Newtonian mechanics (same for all observers). Length is also absolute. These absolute quantities simplify Newtonian mechanics but must be abandoned in special relativity where time, length, and simultaneity become frame-dependent.
\nSolved Example: Airplane in Wind
An airplane flies at 200 km/h in still air. Wind blows at 50 km/h from west to east. (a) Find heading to fly due north. (b) Find ground speed. Solution: (a) To go north, the westward component of plane's velocity must cancel eastward wind. Plane must head west of north by angle θ where 200sinθ = 50, so θ = 14.5°. Heading is 14.5° west of north (345.5° from north). (b) North component: 200cos(14.5°) = 193.6 km/h. Ground speed = 193.6 km/h due north.
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