Relative velocity

Introduction

Relative Velocity is not an absolute quantity, but it is measured relative to an observer. Thus, to measure the velocity of an object, we specify the frame of reference. Ordinarily, we consider the surface of Earth as a frame of reference and determine the velocity of anybody concerning it, that is Earth. Thus, we can say that a car moves at 60 km/hour. It means that the car moves 60 km/hour relative to the Earth’s surface.

Suppose two cars, A and B, are travelling on the straight level road in the same direction,

Va is the Velocity of car A concerning Earth. We want to find the Velocity of car A with respect to car B, which is vector Vab. Here frame of reference is car B.If we add Velocity – Vb to both the cars. Then car B comes to rest. That is, car B and Earth become identical frames of reference. Then the Velocity of car A with respect to car B is Vab which is equal to Va  – Vb.

The direction of relative velocity will depend upon the direction of  Va  – Vb . Thus, to find out the relative velocity of body A with respect to body B in one-dimensional motion, we subtract the velocity of B from both bodies. This effectively brings body B to rest. The resultant Velocity of body A is  Vab , that is, the relative velocity of body A with respect to body B.

Consider two objects, all bodies A and B moving along a straight line in the same direction with uniform velocity Va and Vb, respectively. Xb the displacement from origin O at T is equal to zero Va and Vb are the velocities of body A and body B, respectively. When Va = Vb, the distance between two bodies will remain the same at all times. Their position-time graphs are parallel to each other.

Relative Velocity Mathematical Example

With the help of the following example, we can better understand the notion of relative velocity.

Consider the following scenario: A plane is flying southward at a speed of 100 km/hr. A 25-kilometre-per-hour westerly breeze buffets it. Calculate the plane’s resultant velocity.

Given the wind velocity = Vw = 25 km/hr,

The plane’s speed is equal to Va=100 km/hr.

The plane’s relative velocity with respect to the ground can be expressed as

The angle formed by the wind’s velocity and the plane’s velocity is 90 degrees. The resultant velocity can be calculated using the Pythagorean theorem as follows:

(100 km/hr)2 + (25 km/hr)2 = R2.

100000 km2/hr2 + 625 km2/hr2 = R2

 10 625 km2/hr2  = R2

As a result, R = 103.077 km/hr

Relative Velocity in 1-Dimension

In physics, we consider situations relative to an outside observer, but in the case of relative velocity, the situation measured by an observer inside the problem may be helpful. To explain this concept, let’s take an example of two cars as A and car B, travelling along a motorway, with B travelling at 24 ms-1 and A a distance of 45 m behind it travelling in the same direction at 28 ms-1. The positive direction is defined as how both cars are travelling on the highway. We can utilise the relative velocity of one car to the other to calculate how long it will take the quicker car to catch up to the slower car.

We must subtract the velocity of the vehicle we are witnessing from the velocity of the other car to get the relative velocity of the two cars. We can do this because both velocities are on a straight line, so we don’t have to worry about the directions of the velocities.

Now, if we want to find the velocity of the fastest car, A, relative to the slower car, B, we should take 24 ms-1 away from 28 ms-1 that, is Va – Vb = 4 ms-1 which is in the positive direction the fast car is moving in the direction of the slower car.

To find the velocity of the slower car B relative to the faster car A, we take 28ms-1 away from 24ms-1, Vb-Va = -4ms-1, which is a velocity of 4 ms-1 in the opposite direction to which the cars are driving. The slower vehicle appears to be approaching the quicker vehicle in simple terms.

The velocities of two objects may not necessarily be in a straight line. The same concept of subtracting one velocity from another applies in this scenario, but it must do it vectorially.

Relative Velocity in 2 – Dimensions

Consider two objects, P and Q, moving along parallel lines in the same direction at uniform velocities, v1 and v2 respectively. Time ‘t’ was zero when they began, and their distances from the origin were x01 and x02, respectively.

When the time is changed to ‘t,’ and their displacements are changed to x1 and x2 in relation to the origin with the position axis, 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)          (v2 = vQ , v1 = vP )

The equation is: x01 and x02 are the starting displacements of the item Q with respect to the object P at time

’t = 0′.  x0 = x02 – x01….(4)

Simply replacing the value of eq (4) into (3) yields the following 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:

x – x0t =  (v2 – v1)….(7)

Conclusion

The sense of being aware of oneself moving at a relative velocity to the external world or background is best defined as relative velocity. This can range from being aware that you are travelling at the same speed as the external world around you to being aware that you are accelerating far faster than anything else. Now, you’ve learned the formula to calculate relative velocity along with some examples and applications to help you understand everything more clearly. A healthy understanding of relative velocity will help you get a better sense of its meaning and purpose. Hopefully, you’re excited to put what you’ve learned here into practice.