01 / CHAPTER 09
Relative motion
Translate position, velocity, and acceleration between moving frames.
ANCHOR EQUATION
Relative velocity
In symbols: vA/B = vA − vB
The same event can have different measured velocities in different frames.
Relative motion subtracts descriptions
Velocity is measured relative to a frame. If observer A sees object P move and observer B moves relative to A, P’s velocity relative to B is found by subtracting B’s motion, not by inventing a new force.
Differentiate the position subtraction
Let r_P/B = r_P/A − r_B/A. Differentiating with respect to time in non-relativistic frames gives v_P/B = v_P/A − v_B/A. If both frames have constant relative velocity, acceleration is unchanged: a_P/B = a_P/A. This is the Galilean transformation and assumes speeds far below c.
A boat moves 5.0 m/s north relative to still water while the river flows 2.0 m/s east. Its ground velocity is (2, 5) m/s, speed √29 = 5.39 m/s, and direction 21.8° east of north.
When subtraction needs relativity
At speeds near the speed of light, time and space transform together and velocities do not add this way. Even at ordinary speeds, a rotating or accelerating frame adds apparent-acceleration terms.
Work it: A person walks 1.2 m/s toward the front of a train moving 18 m/s east. Find the person’s velocity relative to the ground and relative to someone walking backward at 0.8 m/s in the train.