Have you ever noticed that the objects outside appear to move backwards when you look out the window? We know that velocity is the speed of an object with its direction. The velocity of an object X in relation to another object Y is called relative velocity. In layman’s terms, it’s the rate at which object X’s relative position changes with respect to object Y.

While travelling on a bus, you must have seen that the people inside the bus are at rest, but things outside the bus keep moving backwards. This is because the object outside moves at a relative velocity regarding the people inside the bus. Let us learn more about the concept of relative velocity and how it is used in this article.

Relative Velocity Definition

The velocity of an object or observer B in the rest of another object or observer A is relative velocity. The following is the general formula for velocity: \(\overrightarrow{v_b\;} -\overrightarrow{v_a\;}\)

Motion

Now let’s discuss everything about motion in detail.

Motion in One Dimension

When a body only moves in a single direction with respect to the chosen axis, we call it motion in one dimension. Only the parameter of one particular direction changes while the rest remains the same.

Example: A ball thrown vertically upwards is a motion in one dimension; the parameter along the horizontal does not change while along the vertical, the velocity and the position of the ball change.

A bike moving on a straight line with some acceleration is also an example of motion in one dimension.

Motion in Two Dimension

When a particle’s kinematic parameters vary in both the horizontal and vertical direction, such a type of motion is known as motion in two dimensions.

Example: A particle is projected at some angle with the horizontal it moves in both the direction, vertical and the horizontal, such type motion is known as motion in two dimensions.

Reference Point

To define the position of a particle, we need a reference point. Without a reference point, we cannot define the position of a particle. Usually, the preference of the reference point is the stationary one. Therefore it is known as the stationary frame of reference. If the reference is non-stationary, then the kinematics parameters will change, and this is where the relative motion comes into play. We use the concept of relative motion to analyse the motion in such cases.

Suppose two cars are moving in the same direction with the same velocities, then the difference between the cars remain constant; that is, if an observer is in the car, it will not see any change in the distance between the cars; therefore, the observer will observe the second car to be stationary, this gives us the idea that any non-stationary observer will observe all the objects moving with the same velocity as the observer with respect to a stationary frame will appear to be at rest. Thus, the observer will always observe itself to be at rest.

Also, suppose the second car has the velocity towards the observer. In that case, it will appear to move towards the observer with double the speed concerning the stationary frame. If the second car moves in the same direction with a slightly greater speed than the observer, the observer will observe the car move at a slower speed. Therefore, we can conclude that the relative velocity is given by subtracting the observer’s velocity from the body’s velocity concerning the stationary frame.

\({\overrightarrow v_{rel}} = {\overrightarrow v_1} – {\overrightarrow v_{{\text{obs}}}}\)

Relative Velocity in One Dimensional Motion

When the observer has kinematics parameters changing only in one direction, and the particle whose kinematic parameters are to be analysed also has motion in the same direction as the observer, we call it relative motion in one dimension. The above example of the cars is an example of relative motion in one-dimensional motion.

Relative Velocity in Two Dimensional Motion

When the observer and the particle have kinematics parameters changing in different directions, the motion is relative motion. For example, A boat moving at an angle with the river’s flow, then the relative velocity of the boat with respect to the river is known as relative velocity in two dimensions.

An aeroplane moving at some angle with the wind then the aeroplane’s motion with respect to the wind is also an example of relative motion in two dimensions.

If a particle is dropped from the aeroplane, then the particle’s motion with respect to the aeroplane is also an example of relative motion in two dimensions.

River Boat Problem

In these type of problems, at first we define the velocity of the stream or river \(\overrightarrow v_R.\)

Let the velocity of the boat relative to ground be \(\overrightarrow v_B.\)

Let the velocity of the boat relative to the river be \(\overrightarrow v_{BR}.\)

Then we have,

\({\overrightarrow v_B} = {\overrightarrow v_{BR}} + {\overrightarrow v_R}\)

Let the angle At which the boatman steers be \(\theta\).

The velocity at which the boats moves in direction perpendicular to the flow of river is given by,

\(\overrightarrow v_{BR} \cos \,\left( \theta \right)\)

The velocity at which the boat moves in the direction of the flow of the river is given by,

\(\overrightarrow v_R – \overrightarrow v_{BR} \sin \,\left( \theta \right)\)

The time taken by the boat man to cross the river will be,

\(t = \frac{d}{{\left| {\overrightarrow v_{BR} \cos \,\left( \theta \right)} \right|}}\)

Drift is the total displacement in the direction of the river’s flow, which is given by the product of the velocity in that direction and the total time taken to cross the river.

\(x = \frac{d}{{\left| {\overrightarrow v_{BR} \cos \,\left( \theta \right)} \right|}}\left( {\overrightarrow v_{R} – \overrightarrow v_{BR} \sin \,\left( \theta \right)} \right)\)

Now, we have some special cases such as,

1. Condition for the boat to cross the river in shortest time.
   We can clearly visualise if angle \(\theta = 0^\circ \), then the velocity of the boat in the direction perpendicular to the direction of the flow of the river is maximum, thereby giving us the minimum time required to cross the river.
   \(\theta = 0^\circ \)
   \( \Rightarrow \cos \left( \theta \right) = 1\)
   \(t = \frac{d}{{\left| {\overrightarrow v_{BR} \cos \,\left( \theta \right)} \right|}}\)
   \( \Rightarrow {t_{\min }} = \frac{d}{{\left| {\overrightarrow v_{BR}} \right|}}\)
2. Condition for zero drift.
   For drift to be zero the velocity in the direction of the flow of the river must be zero
   \(x = 0\)
   \( \Rightarrow x = \frac{d}{{\left| {\overrightarrow v_{BR} \cos \,\left( \theta \right)} \right|}}\left( {\overrightarrow v_{R} – \overrightarrow v_{BR} \sin \,\left( \theta \right)} \right)\)
   \( \Rightarrow \left( {\overrightarrow v_{R} – \overrightarrow v_{BR} \sin \,\left( \theta \right)} \right) = 0\)
   \( \Rightarrow \sin \,\left( \theta \right) = \frac{{\left| {\overrightarrow v_{R}} \right|}}{{\left| {\overrightarrow v_{BR} } \right|}}\)

Rain Umbrella Problem

In these types of problems, we have rain falling at some angle while the man moves with some velocity.

Let the velocity of the rain with respect to the ground be, \({\overrightarrow v_R}.\)

Let the velocity of the man with respect to the ground be, \({\overrightarrow v_M}.\)

Let the velocity of the rain with respect to the man is \({\overrightarrow v_{RM}}.\)

\({\overrightarrow v_{RM}} = {\overrightarrow v_R} – {\overrightarrow v_M} .\)

If the rain is falling at some angle \(\theta\) with respect to the vertical.

Then, the velocity of the rain will have a component in both the horizontal and the vertical directions.

\({\overrightarrow v_R} = v_R \sin (\theta) \widehat i + v_R \cos (\theta) \widehat j\)

\({\overrightarrow v_{RM}} = (v_R \sin (\theta) – v_M) \widehat i + v_R \cos (\theta) \widehat j\)

If we want the rain to appear to fall vertically then, the velocity component of the relative velocity should be zero in the horizontal direction.

\( \Rightarrow \left( {{v_R}\sin\left( \theta \right) – {v_M}} \right) = 0\)

\( \Rightarrow \sin\left( \theta \right) = \frac{{\left| {\overrightarrow v_M} \right|}}{{\left| {\overrightarrow v_R} \right|}}\)

Applications

When we are sitting in a moving train then if we look out through the window, then the objects appear to move and we feel ourselves to be at rest.

Solved Problems

Q.1. A man wishing to cross a river flowing with velocity \(u\) jumps at an angle \(\theta\) with the river flow.
(a) Find the net velocity of the man with respect to the ground if he can swim with speed \(v\) in still water.
(b) In what direction does the man actually move?
(c) Find how far from the point directly opposite to the starting point does the boat reach the opposite bank, if the width of the river is \(d\).
Ans:

The velocity of the river with respect to ground is \(u\).
The velocity of the man in still water is \(v\).
Angle at which the man jump is \(θ\).
Width of the river is \(d\).
Net velocity of the man is resultant velocity,
\({\overrightarrow v _{{\text{net}}}} = \overrightarrow v + \overrightarrow u \)
Magnitude is given by,
\(\sqrt {{v^2} + {u^2} + 2uv\,\cos \left( \theta \right)} \)
The direction of motion is given by,
\(\tan \left( \phi \right) = \frac{{{v_y}}}{{{v_x}}}\)
\(\tan \left( \phi \right) = \frac{{v\,\sin \left( \theta \right)}}{{u + v\,\cos \left( \theta \right)}}\)
Vertical displacement is the product of vertical velocity and time taken by boat to cross the river,
\(d = v_y × t\)
\( \Rightarrow t = \frac{d}{{v\sin \left( \theta \right)}}\)
Putting the value of t the horizontal drift is given by,
\(x = v_x × t\)
\( \Rightarrow x = \left( {u + v\cos \left( \theta \right)} \right)\frac{d}{{v\sin \left( \theta \right)}}\)

Q.2. To a man running upwards on the hill, the rain appears to fall vertically downwards with \(4\,\rm{m}\;\rm{s}^{-1}\). The velocity vector of the man with respect to earth is \(\left( {2\widehat i + 3\widehat j} \right)\,{\text{m}}\,{{\text{s}}^{ – 1}}\) if the man starts running down the hill at the same speed, then determine the relative speed of the rain with respect to man.

Ans: Velocity of rain with respect to man \({\overrightarrow v _{r,m}} = {\overrightarrow v _r} + {\overrightarrow v _m}\),
The direction of rain is negative y-direction with respect to man so resultant can be written as 
\( – 4\widehat j = \overrightarrow v_r – \left( {2\widehat i + 2\widehat j} \right)\)
\(\overrightarrow v_r= 2\widehat i – \widehat j\)
When man starts running down,
\(\overrightarrow v _{r,m} = \overrightarrow v _{r}- \overrightarrow v _{m}\)
\( = 2\widehat i – \widehat j + 2\widehat i + 3\widehat j\)
\( = 4\widehat i + 2\widehat j\)
Now determining the magnitude of the vector, i.e., speed
\( = \left| \overrightarrow v _{r,m} \right| = \sqrt {16 + 4} \)
\( = \sqrt {20} \,{\text{m}}\,{{\text{s}}^{ – 1}}.\)

Summary

1. The relative velocity of a body is given by subtracting the observer’s velocity from the body’s velocity with respect to the stationary frame.
   \({\overrightarrow v_{{\text{rel}}}} = {\overrightarrow v_1} + {\overrightarrow v_{{\text{obs}}}}\)
2. For minimum time to cross the river, the angle at which the boat is steered should be zero that is perpendicular to the river’s flow.
   \( \Rightarrow {t_{min}} = \frac{d}{{\left| \overrightarrow v_{BR} \right|}}\)
   Where,
   The velocity of the boat relative to the river is \(\overrightarrow v_{BR}.\)
   The width of the river is \(d.\)
3. For zero drift, the direction of the net velocity of the boat should be in the direction perpendicular to the flow of the river. That is the component of the boat’s velocity in the direction of the flow of the river should be zero.
   \( \Rightarrow \left( {\overrightarrow v_{R} – \overrightarrow v_{BR} \sin \left( \theta \right)} \right) = 0\)
   \( \Rightarrow sin\left( \theta \right) = \frac{{\left| \overrightarrow v_{R} \right|}}{{\left| \overrightarrow v_{BR} \right|}}\)
   Where,
   The velocity of the stream or river is \(\overrightarrow v_{R} .\)
   The velocity of the boat relative to the river is \(\overrightarrow v_{BR} .\)
4. For rain to fall vertical the horizontal component of the rain, the horizontal component of the velocity of the rain should be zero.
   \( \Rightarrow \sin \left( \theta \right) = \frac{{\left| \overrightarrow v_{M} \right|}}{{\left| \overrightarrow v_{R} \right|}}\)
   Where,
   The velocity of the rain with respect to the ground is, \(\overrightarrow v_{R} .\)
   The velocity of the man with respect to the ground is, \(\overrightarrow v_{M}.\)
5. If nothing is mentioned for a value of the velocity then that quantity is to be considered to be with respect to earth.

FAQs

Q.1. What is the expression for relative velocity?
Ans: The relative velocity of a body is given by subtracting of the observer’s velocity from the velocity of the body with respect to the stationary frame.
\({\overrightarrow v_{{\text{rel}}}} = {\overrightarrow v_1} + {\overrightarrow v_{{\text{obs}}}}\)

Q.2. Are the equations of motion valid for relative motion when the reference is non-stationary?
Ans: Yes the equation of motions are valid for relative velocity but it is important to note that all the values must be from the same frame of reference.

Q.3. What is the condition for boat to cross the river width with minimum time?
Ans: For minimum time to cross the river the angle at which the boat is steered is should be zero that is perpendicular to the flow of the river.
\( \Rightarrow {t_{min}} = \frac{d}{{\left| \overrightarrow v_{BR} \right|}}\)
Where,
The velocity of the boat relative to the river is \(\overrightarrow v_{BR}.\)
The width of the river is \(d.\)

Q.4. What is condition for zero drift for boat moving in a river?
Ans: For zero drift the direction of the net velocity of boat should be in direction perpendicular to the flow of the river that is the component of the velocity of the boat in the direction of the flow of the river should be zero.
\( \Rightarrow \left( {\overrightarrow v_{R} – \overrightarrow v_{BR} \sin \left( \theta \right)} \right) = 0\)
\( \Rightarrow \sin \left( \theta \right) = \frac{{\left| \overrightarrow v_{R} \right|}}{{\left| \overrightarrow v_{BR} \right|}}.\)
Where,
The velocity of the stream or river is \(\overrightarrow v_{R}.\)
The velocity of the boat relative to the river is \(\overrightarrow v_{BR}.\)

Q.5. What is the condition for rain to fall vertically?
Ans: For rain to fall vertical the horizontal component of the rain, the horizontal component of the velocity of the rain should be zero.
\( \Rightarrow \sin \left( \theta \right) = \frac{{\left| \overrightarrow v_{M} \right|}}{{\left| \overrightarrow v_{R} \right|}}\)
Where,
The velocity of the rain with respect to the ground is, \(\overrightarrow v_{R}.\)
The velocity of the man with respect to the ground is, \(\overrightarrow v_{M}.\)

Study About Stokes Law

We hope this article on relative velocity is helpful to you. If you have any queries on this page or in general about relative velocity, ping us through the comment box below and we will get back to you as soon as possible.