Relative motion in Physics

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 have a physics homework problem on relative velocity and you are struggling to solve it, feel free to connect with one of our Physics tutors right away)

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.

Example of relative motion in our 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 the 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 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 plane in a storm

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

The frame of reference and its importance in Relative motion in Physics

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 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 reduce the problems based on relative motion in Physics very easy to solve.

Relative motion analysis: position, velocity, and acceleration

Consider two frame of references ${{\rm{S}}_{\rm{1}}}$ and ${{\rm{S}}_{\rm{2}}}$ and suppose a particle P is observed by both frame of references. 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 ${{\bf{r}}_{P,{S_1}}} = {{\bf{O}}_1}{\bf{P}}$ . This 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 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.

And 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 particle with respect to frame ${{\rm{S}}_{\rm{2}}}$ and

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

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

The formula of relative velocity using vector

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

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 is 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.

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 are 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 calculating with the velocity of A, 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 stationary frame, then from the definition, 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 train and train is moving velocity

${{\bf{V}}_{{\rm{T}},{\rm{G}}}}$ relative to the ground,

Now, 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 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.

(1) Down stream $(\theta = {0^o})$

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 ${t_1} = \frac{d}{{({V_{B,\,R}} + {V_{R,\,G}})}}$

(2) Up stream $(\theta = {180^o})$

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 ${t_2} = \frac{d}{{({V_{B,\,R}} - {V_{R,\,G}})}}$

(3) General approach :

Let the boat starts 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 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 ${V_{R,\,g}} < {V_{B,\,R}}\sin \theta$ .

The motion of the boat crossing the river in the shortest time :

The time taken by 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 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 boat with river flow or bank is $({90^o} + \theta )$ .

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

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Author

Rajesh Kumar

20 years of experience teaching high school and college physics to students across the globe.

When not teaching or mentoring, I write informative articles in physics and related subjects. So far, I have written more than 200 articles on different topics in physics. Apart from physics, I am proficient in engineering statics, dynamics, and calculus. I love spending time with my kids and listening to old Hindi songs.

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