What is relative velocity?

Short Answer:

Relative velocity is the velocity of one object with respect to another moving object. It shows how fast and in what direction one body appears to move when observed from another body that is also in motion. In simple words, relative velocity helps us understand the motion of an object as seen by another moving observer.

For example, if two cars move in the same direction at different speeds, the slower car appears to move backward when viewed from the faster one. Relative velocity is very important in studying motion between two bodies moving in the same or opposite directions.

Detailed Explanation :

Relative Velocity

In mechanics, all motions are measured with respect to a reference point or observer. The concept of relative velocity arises when two or more bodies are moving, and we want to find the velocity of one body as seen from the other. It tells how one object appears to move in relation to another object.

If both bodies are moving in the same line (either in the same or opposite directions), relative velocity can be easily calculated by using vector subtraction. Thus, relative velocity is the vector difference between the velocities of two moving bodies.

Let two objects A and B move with velocities  and  respectively. Then the velocity of A relative to B is given by:

and the velocity of B relative to A is:

It can be noted that , meaning that both have equal magnitudes but opposite directions.

Case 1: When Both Bodies Move in the Same Direction

If two bodies move in the same straight line and in the same direction, their relative velocity is the difference of their velocities.

Example:
Let car A move at 80 km/h and car B at 60 km/h, both in the same direction.
Then,

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

If both cars move at the same speed, the relative velocity between them is zero, meaning they appear stationary to each other.

Case 2: When Both Bodies Move in Opposite Directions

If two bodies move in opposite directions along the same line, their relative velocity is the sum of their velocities.

Example:
If car A moves at 70 km/h to the east and car B moves at 50 km/h to the west,

So, each car appears to approach the other with a speed of 120 km/h.

Case 3: When Both Bodies Move at an Angle

When two objects move in different directions making an angle  between them, the relative velocity is obtained using vector subtraction and Pythagoras’ theorem:

This formula is used when motions are not along the same line (for example, one object moving north and the other east).

Graphical Representation of Relative Velocity

The concept of relative velocity can be clearly understood by using a vector triangle.
If two velocity vectors  and  are drawn from a common point, then the vector drawn from the tip of  to the tip of  represents , i.e., the velocity of A relative to B.

This vector representation helps in finding both the magnitude and direction of relative velocity in two-dimensional motion.

Practical Examples of Relative Velocity

  1. Two Moving Cars:
    If two cars are moving on a highway, one faster than the other, the faster car appears to move away slowly when viewed from the slower car. This is relative velocity in action.
  2. Man Walking in a Moving Train:
    A man walking forward at 2 m/s in a train moving at 10 m/s has a velocity of  m/s relative to the ground, but 2 m/s relative to the train.
  3. Boats in a River:
    A boat moving in a river experiences relative velocity with respect to the water. The effective velocity depends on both the boat’s speed and the river current.

    • When the boat moves along the current, relative velocity = (boat speed + current speed).
    • When it moves against the current, relative velocity = (boat speed – current speed).
  4. Aircraft in Wind:
    The velocity of an airplane relative to the ground depends on both its speed in still air and the velocity of the wind. This is another real example of relative velocity in two dimensions.

Importance of Relative Velocity

  • It helps in analyzing the motion of bodies observed from another moving frame.
  • It is used in navigation of ships and aircrafts to find the correct path in the presence of wind or water current.
  • It is important in collision problems and in studying pursuit and chase situations.
  • Engineers use it to design moving systems like conveyor belts, robotic arms, and vehicles that interact with other moving bodies.
Conclusion

Relative velocity is the measure of how fast and in what direction one body appears to move when viewed from another moving body. It is determined by the vector difference of their velocities. The concept applies to all types of motion—linear, angular, or two-dimensional—and is used widely in mechanics, navigation, and motion analysis. It helps us understand how motion changes when viewed from different reference frames, making it one of the most useful concepts in physics and mechanical engineering.