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We know that velocity is a function of time. It is the speed of an object with its direction. However, when we are talking about the velocity of one object with respect to another that means we are discussing the concept of relative velocity. So, what is relative velocity? Relative velocity is the velocity of an object A with respect to another object B. In simple words, it is the rate of change of relative position of object A with respect to object B.

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Now, let’s analyze relative velocity.

Consider two objects, P and Q, traveling with uniform velocities, v1 and v2 along the parallel lines in the same direction. At the time they started, time ‘t’ was zero, and their displacements from the origin were x01 and x02, respectively.

Now, when the time becomes ‘t’ and their displacements become x1 and x2 with respect to the origin with the position axis, then the equation for object P becomes:

x1 = x01 + v1t…..(1)

Similarly, for object Q, the equation becomes:

x2 = x02 + v2t…..(2)

Subtracting equation (1) from (2), we get:

(x2 - x1) = (x02 - x01) + (v2 - v1)t….(3)

Since x01 and x02 are the initial displacements of the object Q with respect to the object P at time ‘t = 0’, so the equation is:

x0 = x02 - x01….(4)

Now, substituting the value of eq (4) in (3), we get the new equation:

(x2 - x1) = x0 + (v2 - v1)t….(5)

One more thing to note here: (x2 - x1) is the relative displacement of the object Q with respect to the object P at time ‘t’, so we rewrite equation (5) as:

x = x0 + (vQ - vP)t….(6)

Rearranging equation (6) as:

\[\frac{x-x_{0}}{t}\] = (v2 - v1)....(7)

We know that the change in the displacement per unit time is velocity, and the same thing can be observed in equation (7), where LHS equals the RHS.

Relative Velocity for Objects Moving in the Same Direction

Now, the equation for the relative velocity of the object Q with respect to the object P from equation (7) is:

vQP = v2 - v1 ….(8)

Relative Velocity for Objects Moving in the Opposite Direction

Now, the equation for the relative velocity of the object Q with respect to the object P from equation (7) is:

vQP = v2 + v1 ….(9)

The dimension of relative velocity is the same as that of the velocity and it is given by:

M⁰L¹T⁻¹

Now, let’s discuss a few questions on relative velocity.

Question 1: What Would Happen if Both the Objects Travel with the Same Velocity?

Answer: If both the objects have the same velocity, then,

vQP = v2 - v1.

If v2 = v1, then x - x0 = 0, or x = x0, which means these two objects will remain at the constant distance apart, i.e, their relative distance, and therefore, the position-time graph for the same will be parallel lines. The graph for this condition is drawn below.

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Question 2: Consider the Following Two Cases for Relative Velocities and Express them Mathematically.

If v2 < v1

If v2 > v1

Answer:

In the first case, when v1 > v2, the difference between the velocities will be negative; also the difference x - x0 will be negative. It means that the separation between the two objects traveling with respect to each other goes on decreasing by the amount v1 - v2 after each time interval.

In the second case, when v1 < v2, the difference between the velocities will be positive; also the difference x - x0 will be positive. It means that the separation between the two objects traveling with respect to each other goes on increasing by the amount v1 - v2 after each time interval.

The graph for both cases is as follows.

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Now, let’s have a look at a few examples of relative velocity and relative motion.

Example 1:

Consider a boy running with a relative velocity of vrel\[^{\rightarrow}\] on a train running with a velocity of vt\[^{\rightarrow}\] relative to the ground, so the speed of the boy relative to the ground will be:

v\[^{\rightarrow}\] = vrel\[^{\rightarrow}\] + vt\[^{\rightarrow}\]

Example 2:

Consider a woman running on the race track in the direction of her competitors (running with a velocity of vc\[^{\rightarrow}\]) with a velocity of vrel\[^{\rightarrow}\], then the equation for the relative velocity becomes:

v\[^{\rightarrow}\] = vrel\[^{\rightarrow}\] - vc\[^{\rightarrow}\]

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

v\[^{\rightarrow}\] = vrel\[^{\rightarrow}\] + vc\[^{\rightarrow}\]

Example 3:

If a satellite is moving in the equatorial plane with a velocity of sand any point on the earth’s surface with a velocity of erelative to the center of the earth, then the relative velocity of a satellite with respect to the surface of the earth will be:

vse\[^{\rightarrow}\] = vs\[^{\rightarrow}\] - ve\[^{\rightarrow}\]

If this satellite moves from the west to east, i.e., in the direction of the rotation of the earth on its axis, then the equation becomes:

vse = vs - ve

FAQ (Frequently Asked Questions)

Question 1: What is Relative Velocity?

Answer: The relative velocity is the velocity of an observer S in the rest frame of another observer R. The general formula of velocity is given by:

The velocity of the observer S relative to observer R is vs^{→} - vr^{→} . This basic formula describes the concept of relative velocity.

Question 2: Why Do We Study the Relative Motion in One Dimension First?

Answer: We study the concept of relative motion in one dimension first because the velocity vectors, for simplifying it, have two possible directions. We can consider the example of a woman sitting on a train running towards the east.

Question 3: What Do You Mean By a Relative Motion Velocity in a Plane?

Answer: The concept of relative motion velocity in a plane is similar to that of relative velocity in a straight line. Only the difference lies in the cases we consider; for a relative motion in a plane, we consider multiple objects moving in a frame that is non-stationary with respect to another observer or viewer.

Question 4: Is Relative Velocity Valid for 3-dimensional Motion?

Answer: The concept of relative velocity applies to both two-dimensional three-dimensional motions. It is the velocity with which an object appears to move with respect to another object.