Relative motion and reference frames

Motion relative to what?

Position and velocity are not absolute in classical mechanics. They are measured relative to a chosen reference frame. A passenger sitting in a moving train is at rest relative to the train but moving relative to the ground. Both descriptions can be correct.

A reference frame is a coordinate system plus a clock used to describe events. In classical mechanics, time is assumed to be the same in all inertial frames. This assumption works extremely well at speeds much smaller than the speed of light.

Relative position

Suppose object A and object B have position vectors ec{r}_A and ec{r}_B in some frame. The position of A relative to B is

ec{r}_{A/B}=ec{r}_A-ec{r}_B

This vector points from B to A. The order matters. The position of B relative to A is

ec{r}_{B/A}=ec{r}_B-ec{r}*A=-ec{r}*{A/B}

Relative velocity

Differentiate relative position with respect to time:

ec{v}_{A/B}=ec{v}_A-ec{v}_B

This gives the velocity of A as observed from B. If two cars move east at $30 m/s$ and $20 m/s$, the faster car moves at $10 m/s$ east relative to the slower car. If they move toward each other, relative speed is larger.

Galilean transformation

For two inertial frames moving at constant relative velocity ec{V}, classical mechanics uses Galilean transformations. If frame $S'$ moves with velocity ec{V} relative to frame $S$, then positions are related by

ec{r}'=ec{r}-ec{V}t

Velocities are related by

ec{v}'=ec{v}-ec{V}

Accelerations are the same:

ec{a}'=ec{a}

This invariance of acceleration is why Newton's second law has the same form in all inertial frames.

Inertial frames

An inertial frame is one in which Newton's first law holds: an object with no net force moves with constant velocity. A frame moving at constant velocity relative to an inertial frame is also inertial.

A car moving smoothly at constant velocity is approximately an inertial frame. A car speeding up, braking, or turning is not inertial. In non-inertial frames, apparent or fictitious forces may be introduced to explain observed motion.

Example: walking on a train

Suppose a train moves east at $15 m/s$ relative to the ground. A passenger walks east inside the train at $2 m/s$ relative to the train. The passenger's velocity relative to the ground is

$17 m/s$

If the passenger walks west at $2 m/s$ relative to the train, the velocity relative to the ground is

$13 m/s$

This is classical velocity addition.

Boats and airplanes

Relative motion is essential for navigation. A boat crossing a river has velocity relative to the water, while the water has velocity relative to the shore. The boat's velocity relative to the shore is the vector sum:

ec{v}*{boat/shore}=ec{v}*{boat/water}+ec{v}_{water/shore}

Airplane motion in wind works the same way. The plane's velocity relative to the ground is the plane's velocity relative to air plus the wind velocity.

Limits of Galilean relativity

Classical relative motion works for ordinary speeds. At speeds close to the speed of light, Galilean transformations fail and must be replaced by Lorentz transformations from special relativity. Classical mechanics assumes universal time; relativity shows that this is an approximation.

The big idea

Motion depends on reference frame. Relative position and relative velocity are found by vector subtraction. In inertial frames moving at constant relative velocity, classical mechanics uses Galilean transformations, and acceleration remains invariant. This makes Newton's laws consistent across all inertial observers.