Relative Motion in Physics: Velocity, Frames of Reference Explained

Relative motion in physics is a vital concept that students must master. In this article, we will learn what relative velocity is and how to use this concept to solve real-life physics problems. If you are working through problems on this topic, connecting with an online physics tutor can help you build a solid foundation.

Let us consider two cars, A & B moving in the same direction on the road with equal speed. To a person seated in car A, if he were unconscious of his motion, car B would appear to be at rest. The line joining the two cars will always remain constant in magnitude and direction. The velocity of B relative to A or the velocity of A relative to B is zero.

On the other hand, if A is moving at 30 km/hr and B at 40 km/hr in the same direction, B appears to be moving away from A at a rate of 10 km/hr. This represents the velocity of B relative to A.

If, however, B is moving opposite to the direction of A with a velocity of 40 km/hr, for a person in A, B appears to draw away from him at a rate of 70 km/hr. This represents the velocity of B relative to A. Students preparing for standardized exams can also benefit from working with an AP Physics tutor to master these foundational mechanics concepts.

Examples of Relative Motion in Real Life

Example 1: Relative Motion in 1 Dimension

In one dimensional motion, bodies move along a straight line.

Case-1: Bodies are moving in the same direction.

Case-2: Bodies are moving in the opposite direction.

Consider an example of a person sitting in a car moving with 80 km/hr along +ve x-axis then according to the person sitting in the car, the trees, buildings outside the car moving along –ve x-axis with 80 km/hr.

Example 2: Relative Motion in 2 Dimensions

If we have to find the velocity of A with respect to B and body B is at rest, then the velocity of B with respect to A will be in the opposite direction.

Let us assume two bodies, A and B, moving with velocities V_A and V_B with respect to the earth. We have to find the velocity of A with respect to B, so assuming that B is at rest and yields the velocity of B with respect to A in the opposite direction.

${{bf{V}}_{{rm{A}},{rm{B}}}} = {{bf{V}}_{{rm{A}},{rm{g}}}}-{{bf{V}}_{{rm{B}},{rm{g}}}}$

also, ${{bf{V}}_{{rm{A}},{rm{B}}}} = -{{bf{V}}_{{rm{B}},{rm{A}}}}$

Example 3: Relative Velocity of Rain with Respect to a Moving Man

Suppose a man is walking along east with velocity ${{bf{V}}_{rm{m}}}$ and he finds rain is falling vertically on his head. We can draw this with the help of a figure.

${{bf{V}}_{{rm{r}},{rm{g}}}} = {{bf{V}}_{{rm{r}},{rm{m}}}} + {{bf{V}}_{{rm{m}},{rm{g}}}}$

Then direction of velocity of rain with respect to the ground will be $tan theta = frac{{|{v_{m,,g}}|}}{{|{v_{r,,m}}|}}$ as shown in figure.

Example 4: Motion of a Swimmer Across the River

${{bf{V}}_{{rm{man}},{rm{ ground}}}} = {{bf{V}}_{{rm{man}},{rm{ water}}}} + {{bf{V}}_{{rm{water}},{rm{ ground}}}}$

Example 5: Motion of a Plane in a Storm

${{bf{V}}_{{rm{plane}},{rm{ ground}}}} = {{bf{V}}_{{rm{plane}},{rm{ air}}}} + {{bf{V}}_{{rm{air}},{rm{ ground}}}}$

Understanding how these multi-body motion scenarios connect to broader physics principles is also explored in our guide to condensed matter physics for students and parents.

The Frame of Reference and Its Importance in Relative Motion

A frame of reference is a coordinate system plus a time scale. It calculates the positions, velocities, and accelerations of bodies or objects in that frame. Different frames of reference may move relative to one another.

A frame of reference is particularly important when describing an object’s displacement, velocity, and acceleration. Displacement is the change in the position of an object relative to the frame of reference. The frame can be chosen according to the convenience of the situation.

The position r, velocity V and the acceleration a of a particle depend on the frame of reference. We can relate the position, velocity, and acceleration of a particle measured in two different frames of reference. With the help of a frame of reference, we can analyze relative motion in one dimension, river-boat problem, rain-man problem, and describe the motion of a plane in a storm.

Using a good frame of reference can make problems based on relative motion in physics very easy to solve.

Relative Motion Analysis: Position, Velocity, and Acceleration

Consider two frames of references ${{rm{S}}_{rm{1}}}$ and ${{rm{S}}_{rm{2}}}$ and suppose a particle P is observed by both frames of reference. The frames may be moving with respect to each other.

The position vector of particle P with respect to frame ${{rm{S}}_{rm{1}}}$ is ${{bf{r}}_{P,{S_1}}} = {{bf{O}}_1}{bf{P}}$. The position vector of the particle with respect to frame ${{rm{S}}_{rm{2}}}$ is ${{bf{r}}_{P,{S_2}}} = {{bf{O}}_2}{bf{P}}$. The position of frame ${{rm{S}}_{rm{2}}}$ with respect to frame ${{rm{S}}_{rm{1}}}$ is ${{bf{O}}_{rm{1}}}{{bf{O}}_{rm{2}}}$.

It is obtained that

${{bf{O}}_{rm{1}}}{bf{P}} = {{bf{O}}_{rm{1}}}{{bf{O}}_{rm{2}}} + {{bf{O}}_{rm{2}}}{bf{P}}$

${{bf{V}}_{{S_1},{S_2}}}$

$ Rightarrow {{bf{r}}_{P,{S_1}}} = {{bf{r}}_{P,{S_2}}} + {{bf{r}}_{{S_2},{S_1}}}$…(1)

The position of the particle with respect to ${{rm{S}}_{rm{1}}}$ is equal to the position of the particle with respect to ${{rm{S}}_{rm{2}}}$ plus the position of ${{rm{S}}_{rm{2}}}$ with respect to ${{rm{S}}_{rm{1}}}$.

Differentiating equation (1),

$frac{d}{{dt}}{{bf{r}}_{P,{S_1}}} = frac{d}{{dt}}{{bf{r}}_{P,{S_2}}} + frac{d}{{dt}}{{bf{r}}_{{S_2},{S_1}}}$

$ Rightarrow {{bf{V}}_{P,{S_1}}} = {{bf{V}}_{P,{S_2}}} + {{bf{V}}_{{S_2},{S_1}}}$…(2)

Differentiating equation (2) with respect to time again,

$frac{d}{{dt}}{{bf{V}}_{P,{S_1}}} = frac{d}{{dt}}{{bf{V}}_{P,{S_2}}} + frac{d}{{dt}}{{bf{V}}_{{S_2},{S_1}}}$

$ Rightarrow {{bf{a}}_{P,{S_1}}} = {{bf{a}}_{P,{S_2}}} + {{bf{a}}_{{S_2},{S_1}}}$ …(3)

Equation (1) is known as the equation of relative position.

Equation (2) is known as the equation of relative velocity.

Equation (3) is known as the equation of relative acceleration.

${{bf{V}}_{P,{S_1}}}$ is the velocity of the particle with respect to ${{rm{S}}_{rm{1}}}$.

${{bf{V}}_{P,{S_2}}}$ is the velocity of the particle with respect to frame ${{rm{S}}_{rm{2}}}$.

${{bf{V}}_{{S_1},{S_2}}}$ is the velocity of frame ${{rm{S}}_{rm{2}}}$ with respect to frame ${{rm{S}}_{rm{1}}}$.

If you find these derivations challenging, reading about why physics homework takes so long may help you identify where to focus your study time.

How to Find Relative Velocity in Physics

The relative velocity of body A with respect to body B is given by ${{bf{V}}_{{rm{A}},{rm{B}}}} = {{bf{V}}_{{rm{A}},{rm{g}}}}-{{bf{V}}_{{rm{B}},{rm{g}}}}$

The relative velocity of body B with respect to body A is given by ${{bf{V}}_{{rm{B}},{rm{A}}}} = {{bf{V}}_{{rm{B}},{rm{g}}}}-{{bf{V}}_{{rm{A}},{rm{g}}}}$

$|{{bf{V}}_{A,,B}}| = |{{bf{V}}_{B,,A}}| = sqrt {V_{A,,g}^2 + V_{B,,g}^2 + 2|{{bf{V}}_{A,,g}}||-{{bf{V}}_{B,,g}}|cos theta } $

Here, $theta $ is the angle between $left( {{{bf{V}}_{{rm{A}},{rm{g}}}}} right)$ and $left( {-{{bf{V}}_{{rm{B}},{rm{g}}}}} right)$

For two bodies moving in the same direction, the magnitude of relative velocity is equal to the difference in magnitudes of their velocities.

$left| {{{bf{V}}_{{rm{A}},{rm{B}}}}} right| = {{bf{V}}_{{rm{A}},{rm{g}}}}-{{bf{V}}_{{rm{B}},{rm{g}}}} = left| {{{bf{V}}_{{rm{B}},{rm{A}}}}} right| = {{bf{V}}_{{rm{B}},{rm{g}}}}-{{bf{V}}_{{rm{A}},{rm{g}}}}$

For $theta = {0^o},,cos theta = cos {0^o} = 1$

For two bodies moving in opposite directions, the magnitude of relative velocity is equal to the sum of the magnitudes of their velocities.

$left| {{{bf{V}}_{{rm{A}},{rm{B}}}}} right| = left| {{{bf{V}}_{{rm{B}},{rm{A}}}}} right| = {{bf{V}}_{{rm{A}},{rm{g}}}} + {{bf{V}}_{{rm{B}},{rm{g}}}}$

For $theta = {180^o},,cos theta = cos {180^o} = -1$.

For a broader look at what physics courses cover these mechanics topics, see our overview of A-Level Physics and why it is worth the challenge.

The Formula of Relative Velocity Using Vectors

${{bf{V}}_{{rm{A}},{rm{B}}}} = {{bf{V}}_{{rm{A}},{rm{g}}}} + {{bf{V}}_{{rm{g}},{rm{B}}}}$

Or, ${{bf{V}}_{{rm{A}},{rm{B}}}} = {{bf{V}}_{{rm{A}},{rm{g}}}}-{{bf{V}}_{{rm{B}},{rm{g}}}}$ …(i)

Because as we know, ${{bf{V}}_{{rm{B}},{rm{g}}}} = -{{bf{V}}_{{rm{g}},{rm{B}}}}$

Or, ${{bf{V}}_{{rm{A}},{rm{g}}}} = {{bf{V}}_{{rm{A}},{rm{B}}}} + {{bf{V}}_{{rm{B}},{rm{g}}}}$ …(i)

Here, the meaning of ${{bf{V}}_{{rm{A}},{rm{B}}}}$ is the velocity of body A with respect to body B.

${{bf{V}}_{{rm{A}},{rm{g}}}}$ $ Rightarrow $ Velocity of body A with respect to the ground

${{bf{V}}_{{rm{B}},{rm{g}}}}$ $ Rightarrow $ Velocity of body B with respect to the ground

Equation (i) is the formula for relative velocity using vector notations. Students curious about how physics tutoring costs compare across levels can consult this physics tutor cost guide. Those advancing to higher-level coursework may also benefit from working with a tutor for AP Physics 2.

Importance of Relative Velocity in Physics

Relative velocity has played an important role in physics. For the quantitative description and desired computations, appropriate coordinate systems are taken. There has been an equation of transformation from one frame of reference to another.

In the case of Newtonian mechanics, we come across the Galilean transformation. At the same time, we had to switch over to Lorentz transformation when it was found that the velocity of light is independent of the frame of reference considered. This yielded Einstein’s special theory of relativity.

Definition of Relative Velocity

When the distance between two moving points A and B is changing, either in magnitude or in direction, or both, each point is said to possess a velocity relative to the other.

The velocity of one of the moving points, say A, relative to other point B is obtained by combining the velocity of A with the reversed velocity of B. The velocity of A relative to B is the velocity with which A will appear to move to B, if B is reduced to rest.

If velocity of body A is ${{bf{V}}_{rm{A}}}$ and that of B is ${{bf{V}}_{rm{B}}}$ with respect to a stationary frame, then from the definition, the relative velocity of A with respect to B, ${{bf{V}}_{{rm{A}},{rm{B}}}}$ is given by ${{bf{V}}_{{rm{A}},{rm{B}}}} = {{bf{V}}_{{rm{A}},{rm{g}}}}-{{bf{V}}_{{rm{B}},{rm{g}}}}$.

Examples of Relative Velocity in Physics

There are a lot of applications of relative motion in our daily life.

Relative Motion on a Moving Train

If a boy on a train is running with velocity ${{bf{V}}_{{rm{B}},{rm{T}}}}$ relative to the train, and the train is moving with velocity ${{bf{V}}_{{rm{T}},{rm{G}}}}$ relative to the ground, then the velocity of the boy with respect to the ground will be given by

${{bf{V}}_{{rm{B}},{rm{G}}}} = {{bf{V}}_{{rm{B}},{rm{T}}}} + {{bf{V}}_{{rm{T}},{rm{g}}}}$

Rain-Umbrella Concept

If rain is falling with velocity ${{bf{V}}_{rm{R}}}$ and a man moves with a velocity ${{bf{V}}_{rm{M}}}$ with respect to the ground, he will observe the rain falling with a velocity:

${{bf{V}}_{{rm{R}},{rm{M}}}} = {{bf{V}}_{{rm{R}},{rm{G}}}} + {{bf{V}}_{{rm{G}},{rm{M}}}} = {{bf{V}}_{{rm{R}},{rm{G}}}}-{{bf{V}}_{{rm{M}},{rm{G}}}} = {{bf{V}}_{rm{R}}}-{{bf{V}}_{rm{M}}}$

Check this Khan Academy video on the Rain and man problem for a better understanding.

The Motion of a Boat in a River

Boat motion is classified into three categories based on the angle between V_BR and V_R.

${{rm{V}}_{{rm{BR}}}}:$ Velocity of the boat with respect to the river

${{rm{V}}_{rm{R}}}:$ Velocity of the river with respect to the ground.

Down Stream (θ = 0°)

Resultant velocity of the boat ${{bf{V}}_{{rm{B}},{rm{G}}}} = {{bf{V}}_{{rm{B}},{rm{R}}}} + {{bf{V}}_{{rm{R}},{rm{G}}}}$

The time taken for the boat to move a distance d along the direction of flow of water is ${t_1} = frac{d}{{({V_{B,,R}} + {V_{R,,G}})}}$

Up Stream (θ = 180°)

Resultant velocity of boat $ = {{bf{V}}_{{rm{B}},{rm{R}}}}-{{bf{V}}_{{rm{R}},{rm{g}}}}$

The time taken for the boat to move a distance d opposite to the direction of flow of water is ${t_2} = frac{d}{{({V_{B,,R}} – {V_{R,,G}})}}$

General Approach

Let the boat start at point A on one bank with velocity ${{rm{V}}_{{rm{B}},{rm{R}}}}$ and reach the other bank at point D.

The component of the velocity of the boat antiparallel to the flow of water is ${V_{B,,R}}sin theta $.

The component of velocity of a boat perpendicular to the flow of water is ${V_{B,,R}}cos theta $.

So, the time taken by the boat to cross the river is $d = {V_{B,,R}}cos theta ,t,, Rightarrow ,,t = frac{d}{{{V_{B,,R}}cos theta }}$.

Along with the flow of water, the distance traveled by the boat or the drift (x = BD) is

$x = BD = ({V_{R,,g}} – {V_{B,,R}}sin theta )t$.

Or, BD = drift distance $ = ({V_{R,,g}} – {V_{B,,R}}sin theta ) times frac{d}{{{V_{B,R}}cos theta }}$

(a) The boat will reach the other end of the river to the right of B if ${V_{R,,g}} > {V_{B,,R}}sin theta $.

(b) The boat will reach the other end of the river to the left of point B if ${V_{R,,g}} < {V_{B,,R}}sin theta $.

(c) The boat reaches the exactly opposite point on the bank if ${V_{R,,g}} = {V_{B,,R}}sin theta $.

Motion of the Boat Crossing the River in the Shortest Time

The time taken by the boat to cross the river will be minimum if $cos theta = {(cos theta )_{m,r}}$.

$t = frac{d}{{{V_{BR}}cos theta }} Rightarrow {t_{min }} = frac{d}{{{V_{BR}}{{(cos theta )}_{m,r}}}}$

${t_{min }} = frac{d}{{{V_{BR}}}}$.

Where d is the width of the river.

This time is independent of the velocity of river flow.

Velocity of boat with respect to the ground has magnitude, ${V_{B,,g}} = sqrt {V_{B,,R}^2 + V_{R,,g}^2} $.

The direction of resultant velocity is $theta = {tan ^{ – 1}}left( {frac{{{V_{R,,g}}}}{{{V_{B,,R}}}}} right)$.

Motion of a Boat Crossing the River in the Shortest Distance

This will be possible only when ${rm{x}} = {{rm{x}}_{{rm{min}}}}$.

It means: ${V_{R,,G}} – {V_{B,,R}}sin theta = 0$

$ Rightarrow {V_{B,,R}}sin theta = {V_{R,,G}}$

$sin theta = frac{{{V_{R,,G}}}}{{{V_{B,,R}}}}$.

The angle made by the boat with the river flow or bank is $({90^o} + theta )$.

${V_{R,,g}} = sqrt {V_{B,,R}^2 – V_{R,,g}^2} $

Students taking calculus-based mechanics courses can find additional support from an AP Physics C tutor who specialises in these advanced topics. For those also studying chemistry alongside physics, working with a chemistry teacher online can help balance the workload across science subjects.

Related Reading

• AP Physics 1 Fluids: Is It Easier Per Question?
• 7 Smart Ways to Use Predicted Papers Without Risking Your A-Level Physics Grade
• A-Level Physics: A Blueprint for 2026 Exam Traps Fixed
• AP Physics 1 Exam Prep 2026: Advanced Mastering Mechanics and Circuits