Short Answer:

Relative velocity is the velocity of one object as observed from another object that may also be moving. It describes how fast one body appears to move with respect to another body. In simple terms, it is the difference between the velocities of two bodies.

For example, if two cars move on the same road, one at 80 km/h and the other at 60 km/h in the same direction, then the car moving at 80 km/h appears to move at 20 km/h relative to the slower one. Thus, relative velocity helps to understand motion between two moving bodies.

Detailed Explanation:

Relative Velocity

In mechanics, motion is always observed with respect to some reference point or another body. The relative velocity of one body with respect to another means how fast and in what direction one object appears to move when seen from another moving or stationary object. It is an important concept used in kinematics and dynamics to describe motion between two objects in relative terms.

1. Definition of Relative Velocity

Relative velocity can be defined as the velocity of one body measured from another body. It tells how one object is moving with respect to another.

If two bodies, A and B, are moving with velocities  and  respectively, then the velocity of A relative to B is given by:

Similarly, the velocity of B relative to A is:

It can be noticed that,

which means if A appears to move in one direction with a certain speed relative to B, then B will appear to move in the opposite direction with the same speed relative to A.

2. Motion in the Same Direction

When two bodies are moving in the same direction, their relative velocity is the difference between their velocities.

Example:
If car A is moving at 80 km/h and car B is moving at 60 km/h in the same direction, then the velocity of A relative to B is:

This means car A appears to move at 20 km/h with respect to car B.

In this case, the faster object seems to move slowly relative to the slower one because both are moving in the same direction.

3. Motion in Opposite Directions

When two bodies are moving in opposite directions, their relative velocity is the sum of their individual velocities.

Example:
If car A is moving at 80 km/h east and car B is moving at 60 km/h west, then:

This means the two cars appear to approach each other at a relative speed of 140 km/h.

Thus, when two objects move in opposite directions, they seem to approach or separate much faster than in the same direction.

4. Relative Velocity in Two Dimensions

In two-dimensional motion, relative velocity is determined by vector subtraction of the velocities of the two objects.

If two objects A and B move with velocities  and  at some angle, the relative velocity is found using the vector relation:

The magnitude of relative velocity can be found using the formula:

where  is the angle between their directions of motion.

This form of relative velocity is commonly used in problems related to ships crossing a river, airplanes flying in wind, or particles moving in different directions.

5. Example in Real Life

A very simple example of relative velocity is seen when you are sitting in a moving train. If another train moves beside you at the same speed and in the same direction, it appears stationary to you. This is because the relative velocity between the two trains is zero.

But if the other train moves in the opposite direction, it appears to move very fast relative to you because the relative velocity becomes the sum of both train speeds.

Similarly, in aviation and marine navigation, the concept of relative velocity is used to calculate the effect of wind or water current on the motion of aircraft and ships.

6. Importance of Relative Velocity

• It helps to understand motion between two moving objects.
• It is useful in solving problems in kinematics involving more than one moving body.
• It is applied in navigation, traffic analysis, and mechanical systems involving gears or linkages.
• Engineers use it to analyze the movement of components in machines that interact with each other.

By understanding relative velocity, we can calculate how one object moves concerning another, regardless of the observer’s position.

Conclusion:

Relative velocity is the measure of how fast and in what direction one object appears to move when seen from another object, whether moving or stationary. It is calculated by taking the difference of their velocities as vectors. This concept plays an important role in mechanical engineering, physics, and transportation systems because it helps to describe and analyze motion between two bodies.