Relative Motion

Motion Depends on the Observer

Motion is always described relative to a frame of reference. An object moving quickly past an observer beside the road can have a smaller, zero, or oppositely directed velocity relative to another moving observer.

For ordinary vehicle speeds and parallel, nonrotating axes, we use the Galilean velocity rule. If both velocities are first measured relative to the same ground frame $G$ , the velocity of object $X$ relative to observer $Y$ is:

\vec{v}_{X/Y}=\vec{v}_{X/G}-\vec{v}_{Y/G}

The positive and negative signs record direction along the chosen axis.

The reference frame changes the measured relative velocity. The physical event remains the same.

Relative Motion of Two Cars

Choose the direction of car

B

to see its motion as observed from car

A

Car $A$: $+10\text{ m/s}$
Car $B$: $+24\text{ m/s}$
Relative velocity of $B$ to $A$: $v_{B/A}=+14\text{ m/s}$
As seen from $A$: $B$ appears to move right relative to $A$ .

Observer Velocity Changes Relative Velocity

An object's velocity according to an observer changes when the observer is also moving. The calculation must compare the object's motion with the observer's motion.

Suppose rightward is chosen as positive. A car moves at $+12 \text{ m/s}$ and a taxi moves at $+14 \text{ m/s}$ . To a passenger in the taxi, the car appears to move backward:

v_{\text{car/taxi}}=12-14=-2 \text{ m/s}

The negative value means the relative motion points left.

When two vehicles move in the same direction, the faster observer can see the other vehicle as moving backward. The car still moves forward relative to the road, but its velocity relative to the taxi is negative.

When the Observer Is Still

If the observer is stationary relative to the ground, relative velocity is the same as velocity relative to the ground.

\begin{aligned} &v_{\text{object/observer}} = v_{\text{object/ground}}-0 \\ &= v_{\text{object/ground}} \end{aligned}

That is why, in many simple problems, a velocity with no extra frame stated is usually assumed to be relative to the ground.

What matters is whether the observer is moving. If the observer is stationary, there is no observer velocity to subtract.

When Both Are Moving

If two objects move in opposite directions, their relative speed can be greater than either object's speed relative to the ground. For example, a car moves right at $+12 \text{ m/s}$ , while a truck moves left at $-8 \text{ m/s}$ .

To the truck driver, the car's velocity is:

v_{\text{car/truck}}=12-(-8)=+20 \text{ m/s}

The value $+20 \text{ m/s}$ means the car's position relative to the truck changes rightward at a relative speed of $20\text{ m/s}$ . Whether the vehicles are approaching or separating also depends on which one is initially to the left.

Different observers assign different relative velocities to the same physical motion.

Relative Velocity Is a Difference

Situation	Calculation	Meaning of the relative motion
Observer is still	$v_Y=0$	$v_{X/Y}=v_X$
Observer is faster in the same direction	$12-14=-2 \text{ m/s}$	The object appears to move in the negative direction
Observer moves in the opposite direction	$12-(-8)=+20 \text{ m/s}$	The object appears to move in the positive direction

So, before saying an object is fast, slow, still, or moving backward, always check the frame of reference.