2.1.3.4. Relative Motion

Table of Contents

Seeing Motion from Different Observers

Relative motion means that the description of motion depends on who is observing it. An object can appear to move one way to one observer and another way to a different observer. This does not mean the motion is mysterious. It means that position, velocity, and sometimes acceleration must be described relative to a chosen reference frame.

If you sit inside a moving train and toss a ball straight up, the ball seems to move vertically to you. But a person standing on the ground sees the ball move forward as well as upward and downward. Both descriptions are correct, because each observer measures the motion relative to a different frame.

In this chapter, the main idea is how to relate the motion seen in one frame to the motion seen in another frame.

Relative Position

Suppose there are two objects, A and B. The position of A relative to B tells us where A is when B is used as the reference point.

If $\vec r_A$ and $\vec r_B$ are their position vectors measured from the same origin, then the position of A relative to B is

$$
\vec r_{A/B} = \vec r_A - \vec r_B
$$

This vector points from B to A. In words, to find where A is relative to B, subtract the position of B from the position of A.

If two cars are traveling along a straight road, and car A is at $120 \, \text{m}$ while car B is at $80 \, \text{m}$, then the position of A relative to B is

$$
x_{A/B} = 120 - 80 = 40 \, \text{m}
$$

So A is $40 \, \text{m}$ ahead of B.

Important rule:
$$
\vec r_{A/B} = \vec r_A - \vec r_B
$$
Relative position is found by subtracting the reference object's position from the object's position.

Relative Velocity

Relative velocity tells us how fast one object moves as seen from another object.

If $\vec v_A$ and $\vec v_B$ are the velocities of A and B measured in the same frame, then the velocity of A relative to B is

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

This is one of the most important formulas in kinematics.

For motion in one dimension, signs matter. Suppose car A moves at $25 \, \text{m/s}$ east and car B moves at $15 \, \text{m/s}$ east. Then

$$
v_{A/B} = 25 - 15 = 10 \, \text{m/s}
$$

So A moves away from B at $10 \, \text{m/s}$.

If instead car B moves at $15 \, \text{m/s}$ west, and east is positive, then $v_B = -15 \, \text{m/s}$. So

$$
v_{A/B} = 25 - (-15) = 40 \, \text{m/s}
$$

Now A and B approach or separate much faster because they move in opposite directions.

Important rule:
$$
\vec v_{A/B} = \vec v_A - \vec v_B
$$
Relative velocity is the velocity of one object minus the velocity of the observer or reference object.

Relative Acceleration

Acceleration can also be described relatively. If two objects have accelerations $\vec a_A$ and $\vec a_B$ in the same frame, then the acceleration of A relative to B is

$$
\vec a_{A/B} = \vec a_A - \vec a_B
$$

If the reference frame moves at constant velocity, then its acceleration is zero, and both observers measure the same acceleration.

This is why in many everyday situations, different observers moving steadily relative to each other agree about acceleration.

Important rule:
$$
\vec a_{A/B} = \vec a_A - \vec a_B
$$
If the reference frame has zero acceleration, then relative acceleration equals the ordinary acceleration seen in that frame.

Relative Motion in One Dimension

In one dimensional motion, relative motion is handled with signed quantities. The most important step is choosing a positive direction.

The table below shows common situations.

Situation	Formula	Result
Both move in same direction	$v_{A/B} = v_A - v_B$	Smaller difference
Move in opposite directions	$v_{A/B} = v_A - (-v_B)$	Speeds add
Reference object faster than object	$v_{A/B} < 0$	A appears to move backward relative to B

A negative relative velocity does not mean something is wrong. It simply means the direction is opposite to the chosen positive direction.

Relative Motion in Two Dimensions

In two dimensions, relative motion must be treated as vector subtraction. This means subtracting components.

$$
\vec v_A = v_{Ax}\hat i + v_{Ay}\hat j
$$

and

$$
\vec v_B = v_{Bx}\hat i + v_{By}\hat j
$$

then

$$
\vec v_{A/B} = (v_{Ax} - v_{Bx})\hat i + (v_{Ay} - v_{By})\hat j
$$

The same idea works for relative position and relative acceleration.

This is especially useful in navigation problems, such as boats crossing rivers or airplanes flying in wind. The motion of the boat relative to the water combines with the motion of the water relative to the ground. The motion seen from the ground is found by vector addition, and the relative quantities are found by subtraction.

Frame Transformation Idea

A simple change of reference frame can be written clearly. Suppose frame $S'$ moves with constant velocity $\vec V$ relative to frame $S$. Then for an object with velocity $\vec v$ in frame $S$, its velocity in frame $S'$ is

$$
\vec v' = \vec v - \vec V
$$

This says that to describe motion from the moving frame, subtract the velocity of the moving frame itself.

For position, if the origins coincide at $t = 0$, then

$$
\vec r' = \vec r - \vec V t
$$

This is the basic classical rule for changing from one steadily moving frame to another.

For frames moving with constant relative velocity $\vec V$:
$$
\vec r' = \vec r - \vec V t
$$
$$
\vec v' = \vec v - \vec V
$$
$$
\vec a' = \vec a
$$
These are the classical transformation rules for relative motion.

Example, Boat Crossing a River

Imagine a river flowing east at $3 \, \text{m/s}$. A boat moves north relative to the water at $4 \, \text{m/s}$.

To someone on the shore, the boat has both northward and eastward motion. Its velocity relative to the shore is

$$
\vec v_{\text{boat/shore}} = \vec v_{\text{boat/water}} + \vec v_{\text{water/shore}}
$$

If east is the $x$ direction and north is the $y$ direction, then

$$
\vec v_{\text{boat/water}} = 4\hat j
$$

$$
\vec v_{\text{water/shore}} = 3\hat i
$$

$$
\vec v_{\text{boat/shore}} = 3\hat i + 4\hat j
$$

Its speed relative to the shore is

$$
|\vec v| = \sqrt{3^2 + 4^2} = 5 \, \text{m/s}
$$

and its direction is diagonally across the river.

This is a classic relative motion situation. The boat's own motion is relative to the water, but the observer on shore sees the combination of boat motion and water motion.

Rendering scientific sketch…

Boat crossing a flowing river

Example, Airplane and Wind

An airplane may have a velocity relative to the air, while the air itself moves relative to the ground. Then the airplane's velocity relative to the ground is

$$
\vec v_{\text{plane/ground}} = \vec v_{\text{plane/air}} + \vec v_{\text{air/ground}}
$$

If the wind blows sideways, the plane may drift unless it points slightly into the wind. This is another important application of relative motion.

The pattern is always the same. To connect three objects or frames, keep track of what is moving relative to what.

A Useful Relationship Between Three Frames

Suppose object A moves relative to B, and B moves relative to C. Then the motion of A relative to C is

$$
\vec v_{A/C} = \vec v_{A/B} + \vec v_{B/C}
$$

This relation is extremely useful in practical problems.

For example, boat relative to shore can be found from boat relative to water plus water relative to shore. The same idea works for planes, conveyor belts, moving walkways, and passengers inside vehicles.

Chain rule for relative velocities:
$$
\vec v_{A/C} = \vec v_{A/B} + \vec v_{B/C}
$$
Be careful with the order of subscripts. They matter.

Why Subscripts Matter

Relative motion formulas are simple, but mistakes happen when subscripts are mixed up. The notation $\vec v_{A/B}$ means velocity of A as seen by B. The order is important.

A good habit is to read the fraction-like notation in words. For example,

$$
\vec v_{\text{boat/shore}}
$$

means "velocity of the boat relative to the shore."

Then

$$
\vec v_{\text{boat/shore}} = \vec v_{\text{boat/water}} + \vec v_{\text{water/shore}}
$$

reads naturally and helps prevent errors.

What Relative Motion Does Not Mean

Relative motion in classical mechanics does not mean that all measurements are completely arbitrary. Observers in different frames can convert their measurements using clear rules. Also, this chapter deals with ordinary speeds much smaller than the speed of light. At very high speeds, the transformation rules are different and belong to relativity.

Here, the key point is much simpler. Motion must always be stated relative to some frame, and different frames are related by vector equations.

Final Idea

Relative motion is about comparing motions from different viewpoints. The same object can have different positions and velocities for different observers, but these descriptions are connected by simple subtraction or addition of vectors.

Core formulas of relative motion:
$$
\vec r_{A/B} = \vec r_A - \vec r_B
$$
$$
\vec v_{A/B} = \vec v_A - \vec v_B
$$
$$
\vec a_{A/B} = \vec a_A - \vec a_B
$$
For three frames:
$$
\vec v_{A/C} = \vec v_{A/B} + \vec v_{B/C}
$$

Once the reference frame is chosen carefully and the directions are defined clearly, relative motion becomes a direct and powerful tool for solving real motion problems.

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