Relative Velocity

On occasion, one or even more objects move in a frame that is non-stationary concerning another observer, and we must deal with this situation. For example, a boat crossing a river flowing at a certain velocity or an aeroplane flying through the air while experiencing wind. In all of these instances, we must consider the medium’s influence on the item to depict the object’s total motion accurately. During this process, we determine the relative velocity of the object by taking the velocity of the particle into account and the velocity of the surrounding medium. We will learn how to choose the relative velocity in this section.

Relative Velocity: Definition

The velocity of an object A concerning another object B is referred to as its relative velocity. Put another way. It is the rate at which the close location of object A concerning object B changes.

Let’s look into relative velocity in more detail.

Calculation of 2D Relative Velocity

Think about two objects, P and Q, going in the same direction at uniform velocities, v1

 and v2, along two parallel lines at the same speed. Time ‘t’ was zero at the time of their start, and their displacements from the origin were x01 and x02, respectively, when they began. After all of this, if the time is changed to ‘t’ and the objects’ displacements are changed to x1 and x2 relative to the origin with the position axis, the equation for object P is as follows:

x1 = x01 + v1t …..(1)

In a similar vein, the equation for object Q is as follows:

x2 = x02 + v2t …..(2)

When we subtract equation (1) from equation (2), we get:

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

The initial displacements of the item Q concerning the object P at time ‘t = 0’ are represented by the variables x01 and x02, resulting in the following equation:

x0 = x02 – x01 …..(4)

The following new equation is obtained by putting the value of equation (4) into equation (3):

(x2 – x1) = x0 + (v2 – v1)t …..(5)

Just one more point to mention: the relative displacement between the objects Q and P at time t is represented by x = ( x2 –  x1 ), which means that equation (5) can be written as follows:

x = x0 + (v2 – v1)t …..(6)

Equation (6) can be rewritten as follows:

x-xₒt = (v2 – v1) …..(7)

In our last discussion, we learned that velocity is defined as the change in displacement per unit of time. The same can be seen in equation (7), where the LHS equals the RHS.

Relative Velocity: Moving in the Same Direction

Consider two objects, A and B, travelling with velocities Va and Vb concerning a shared stationary frame of references, such as the ground, a bridge, or a platform that is not moving.

The relative velocity of object A concerning object B can be expressed as follows:

Vab = Va – Vb

Similarly, the velocity of object B concerning object a can be calculated as follows:

Vba = Vb – Va

As we can see from the two expressions above, we can say

Vab = -Vba 

Even though the magnitudes of both relative velocities are identical to one another. Mathematically,

|Vab| = |Vba|

Some Problems Related to Relative Velocity

Question. An oncoming car A  travelling at speed 110 km/h on the road passes another car B moving at 80 km/h in the opposite direction. What is the car A’s velocity as seen by a passenger in car B?

Ans. The car A’s velocity is represented by the letter Va, and the car B’s velocity is represented by the letter Vb.

In this case, the velocity of the motorcycle as seen by a passenger in the car is denoted by the expression.

Vab  = Va – Vb 

Substituting the values in the previous equation results in the following:

Vab = 110 km/h – 80 km/h = 30 km/h.

As a result, the car A  travels at a speed of 30 kilometres per hour compared to the other  car’s passenger.

Question. An athlete who swims across the surface of a river that is moving at the rate of 4 m/s does it with the speed of 2 m/s. Calculate the real swimming velocity of the swimmer as well as the angle.

Ans. The following equation can be used to calculate the real swimming velocity of the swimmer:

Vactual = 2² +4² = 4.47m/s

The angle is calculated as follows:

tanθ = 24

θ = tan-1 24 = 26.57°

Question. Suppose both objects are travelling at the same speed. What would happen?

Ans. If the velocities of both objects are the same, then v2 = v1  (here, x – x0  = 0, or x = x0 ), the distance between the two objects will remain constant (i.e., their relative distance will remain constant), and the position-time graph for the same will consist of parallel lines.

Summary

One or even more objects move in a frame that is non-stationary concerning another observer, and we must deal with this situation. The velocity of an object A with regard to another object B is referred to as its relative velocity. To put it another way, it is the rate at which the relative location of object A with regard to object B changes. We determine the relative velocity of the object by taking the velocity of the particle into account, as well as the velocity of the surrounding medium.