Relative Motion

Relative Motion is defined as the motion of an object when observed with respect to another object which may be either at rest or in motion. The concept of motion is relative in nature rather than absolute. As per the definition of motion, a body is said to be in motion or at rest if it changes its position with respect to the observer or a stationary object. This can be understood from the following example

Example of Relative Motion: Suppose you and your friend are traveling on a train sitting together and a boy standing alongside the track is observing you. Both of you don’t change your position with respect to each other hence both of you are at rest with respect to each other but with respect to the boy who was standing outside you and your friend are in motion because your position is changing with time.

Hence we observe that the state of motion is different for different observers making motion a relative concept rather. The difference in the observation of the boy standing outside and you sitting inside is because of the different Frames of Reference.

Relative Motion Definition

When the motion of an object is observed with respect to some other object either at rest or in motion, i.e., either a moving or stationary object, then the motion of the first object is called relative motion with respect to the second object. 

Since in real life, we are observing all the objects on the Earth’s surface, therefore, the motion of the object is not calculated in reference to the surface of the Earth.

The assumption while computing relative motion is the understanding that the other object is in a static state. Relative motion encompasses all aspects of motion, that is, velocity, speed, or acceleration. 

For example, if we consider a person sitting in an airplane, the person appears to be at a zero velocity relative to the airplane but actually moves with the same velocity as that of the plane taking into account the ground. In this case, the motion is dependent on the observer’s frame of reference. The objects may be moving in the same or different directions with reference to each other. 

Since the objects are in motion when we need to learn the concept of Relative Velocity to know how fast or slow one body is moving.

Relative Velocity

The relative velocity of the body comes into consideration when the other object moves either in the same or opposite direction. The speeds of the involved objects may be increasing, decreasing, or constant with reference to each other. 

Let us assume the initial position of two objects A and B are both at origin, at points x_A (0) and x_B (0) respectively. The corresponding positions of these objects at time instance t will be equivalent to,

x_A (t) = x_A(0) + v_At

x_B (t) = x_B(0) + v_Bt

Displacement from object A to B is given by, 

x_B(t) – x_A(t) = [x_B (0) – x_A (0)] + (v_B -v_A)t

The velocity of B relative to A is given by, 

v_BA = v_B – v_A

The velocity of A relative to B is given by, 

v_AB = v_A – v_B

The concept of relative velocity can be understood by following Position-Time graphs.

Position-Time Graphs Depicting Relative Velocity

The concept of relative velocity can be explained by the Position-Time graph using the following three cases.

Case 1: Two objects moving in the same direction with equal velocities. 

In the above graph, the two objects will appear at rest with respect to one another.

Case 2: Two objects moving with different velocities in the same direction. 

In the above graph,

• The magnitude of V_BA and V_AB will be lower than the magnitude of V_A and V_B. Object A appears faster than B.
• The velocities of the object will not be so large with respect to one another as compared to a stationary environment.

Case 3: Two objects moving with different velocities but in opposite directions. 

In the above graph,

• The magnitude of V_BA and V_AB will be higher than the magnitude of V_A and V_B. 
• Both objects will appear moving faster than one another.

Reference Frames

The motion of an object is always considered with the reference to a time and position frame called as reference frame. The object takes a uniform or non uniform velocity with a particularly visible frame of reference. The frame of reference is considered to be the surface of the earth for all practical purposes. 

For instance, in consideration of the movement of the objects, the surface of the earth is assumed to be the frame of reference and all the motion is considered with reference to the static point taken on the earth’s surface. The objects may be either in the same or opposite direction. Relative motion can be considered either along a straight line or in a plane, in multiple dimensions. 

Motion in One Dimension

Motion in one dimension or straight line refers to the displacement of an object with respect to time while the object moves along a straight path. This type of motion is also called linear motion. It is a uni-dimensional motion and can be well expressed using the X-axis coordinate or Y-axis system alone I.e. position is a function of either the x-coordinate or y-coordinate. For instance, The car moving along the same path as shown below:

Relative Motion in One Dimension

In the case of relative motion in one dimension, the two objects must be in motion along the same axis but the motion and the velocity of the two objects can be either in the same or opposite direction. Hence, the following two cases arise:

Case 1: Objects may be moving in the same direction with reference to each other. 

If the two are moving in the same direction, for instance, a truck and a person with the surface of Earth as the reference frame. The velocity of the train with respect to the person can be written as:

[Tex]\vec{v}_{TP}= \vec{v}_{TE}-\vec{v}_{PE}\space m/s [/Tex]

This speed can be considered to be positive as the speed of the train is greater than the person’s.

Case 2: Objects may be moving in opposite directions with reference to each other. 

If the object and the person are moving in the same direction, for instance, a truck and a person moving in opposite directions with respect to one another, and assume the direction of motion of the train is negative then the relative velocity of the person with respect to the train is given by

[Tex]\vec{v}_{PT}=\vec{v}_{PE}-(\vec{-v}_{ET})\space m/s [/Tex]

[Tex]\vec{v}_{PT}=\vec{v}_{PE}+\vec{v}_{ET}\space m/s [/Tex]

In the above equation, the summation of the velocity vectors can be computed to find the velocity of the person with respect to the train.

The ordering of subscripts can be arranged as per the need i.e. you need to find whose relative velocity compared to whom. For example, if we need to calculate the velocity of the person relative to the earth assuming the train as a reference then the following equation can be used. 

The above equation can be realized by the following example:

Figure 2

Motion in Two Dimensions

It is defined as the motion in which the position of an object is given by a pair of coordinates i.e. x and y. When a body will move in the plane the coordinates of the position will change. For example, if a body is moving in an X-Y plane then the position of the body is given as \vec{r} = x\hat{i}+y\hat{j}

Relative Motion in Two Dimensions

Consider two objects P and Q moving in a plane with positions X_P(x₁,y₁) and X_Q(x_2,y₂). The velocities of these two objects can be determined by differentiating the position vector with respect to time ‘t’.

Let the velocities so obtained for the particles be V_P and V_Q. Then there can be two cases, particles P and Q moving in the same and the opposite direction.

Case 1: When particles moving in the same direction

From the vector law of addition

[Tex]\vec{v}_{P}+\vec{v}_{PQ}=\vec{v}_{Q} [/Tex]

[Tex]\vec{v}_{PQ}=\vec{v}_{Q}-\vec{v}_{p} [/Tex]

Thus, in the case of the motion of particles in the same direction subtract the velocities to find the relative velocity.

Case 2: When particles are moving in the opposite direction

From the vector law of addition

\[Tex]\vec{v}_{PQ}=\vec{v}_{P}+\vec{v}_{Q} [/Tex]

Thus in the case of the motion of particles in opposite directions, the relative velocity is given by the summation of individual velocities.

Also, Read

• Relative Motion in Two Dimension
• Motion in Three Dimension
• Measuring Rate of Motion

Relative Motion Problems

Problem 1: Athletes are participating in a relay race on the track, running with respective velocities of [Tex]\vec v_c           [/Tex] and [Tex]\vec v_{rel}           [/Tex]. Compute the relative velocity. 

Solution:

[Tex]\vec v=\vec vrel-\vec vc [/Tex]

Now, if she moves in the opposite direction, then the equation will be:

[Tex]\vec v=\vec vrel+\vec vc [/Tex]

Problem 2: Consider a satellite moving along the equatorial plane with a velocity of s m/s. Find the relative velocity of the corresponding satellite with respect to the reference point to be the surface of the earth.

Solution:

[Tex]\vec v_{se}=\vec v_s-\vec v_e [/Tex]

Consider the satellite to be moving in the direction of the rotation of the earth on its axis, therefore, the velocity becomes:

[Tex]v_{se}=v_s-v_e [/Tex]

Problem 3: The elevator is moving in an upward direction with a uniform acceleration ‘a’ m/s². The man throws a rubber ball in the upward direction with a velocity ‘v’ relative to the lift. The man catches the ball after a time instance of t seconds. Show that a + g = 2v/t.

Solution:

The lift frame can be assumed to be the point of reference. Therefore, the aspects of motion, acceleration, displacement, and velocity will be considered from the point of reference. When the ball returns to the man, therefore, the displacement from the lift frame becomes zero. Let us assume the velocity of the object with respect to the lift frame is v.

g – (-a) = a + g (↓)   downwards

Now, [Tex]s = ut + \frac{1}{2}at^2                  [/Tex]   

⇒ [Tex]0 = vt – \frac{1}{2} (a+g)t^2 [/Tex]

a + g = 2(v/t) .

Problem 4: The two trains move with different velocities, t₁ with 10 m/s and t₂ with 15 m/s on parallel tracks in reference to each other. Compute the relative velocity of train t₁ with respect to t₂.

Solution: 

Given, 

• v₁ = 10 ms^-1
• v₂ = 15 ms^-1

Relative velocity of slow train w.r.t. the fast train = v₁ – v₂ = 10 – 15 = – 5 ms^-1 

Negative sign shows that slow train appears to move westward w.r.t. fast train with a velocity of 5 ms^-1.

Problem 5: Cart A is moving with a velocity of 40 ms^-1 from North to South along one track, while cart B is moving in an opposite direction to the previous card with a speed of 30 ms^-1 from South to North. Calculate the relative velocities of both the carts with each other. 

Solution: 

Let us consider the direction from North to South to be positive.

Therefore,

• v_A = +40 ms^-1
• v_B = -30 ms^-1

(i) Relative velocity of B w.r.t. A = v_B – v_A = -30 – 40 = – 70 ms^-1 

Therefore, cart B appears to move from South to North with a speed of 70 ms^-1 for any person sitting in cart A.

(ii) Velocity of the ground, v_B = 0, since it is a stationary object. 

Relative velocity of ground w.rt. A = v_B – v_A = 0 – 40 = – 40 ms^-1 

Therefore, the ground appears to move from the south to north direction with a speed of 40 ms^-1 w.r.t. cart A.

Relative Motion – FAQs

What is Relative Motion?

The motion of an object with respect to another object which may be in rest or motion is called Relative Motion.

What is Reference Frame?

Reference Frame is referred to the environment which has some set coordinates according to which changes in the physical property is understood and such change is calculated in relation to the coordinates of Reference Frames.

What is the Difference between Absolute Motion and Relative Motion?

Absolute Motion is when an object changes its position with respect to a body which is at an absolute rest i.e. a tree or a building while Relative Motion is when an object changes its position with respect to a body which may be at rest or motion.

What is Motion in One Dimension?

When a body moves along a line then only one coordinate of its position keeps changing while the rest coordinate remains constant, this is called Motion in One-Dimension.

What is Motion in Three Dimensions?

When a body moves in space then all the 3 coordinates of its position keep changing, this is called Motion in Three Dimensions.

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