Relative Motion JEE Physics

Relative Velocity and Relative Acceleration for IIT JEE

Relative generally means the comparison between any two objects regarding their different characteristics. Relative motion can be defined as the comparison between the motions of a single object to the motion of another object moving with the same velocity. Relative motion can be easily found out with the help of the concept of relative velocity, relative acceleration or relative speed. An object is in a static state when the relative motion of the object with regard to another object which has the relative velocity with the earth, but the relative motion of the object remains constant. Basically, relative motion includes the comparison of relative acceleration of the rigid bodies and the relative velocities of the two objects. We can simply say that the comparisons between the motions of any two objects are called relative motion.

Objects show relative motion with regard to another object which is known as reference frames. When the relative motion of an object like a train, is calculated then the relative motion can be provided with respect to the relative motion of the earth. So in this example, a train is moving with some relative velocity relative to the earth (which is the reference frame of the given example). Reference frames are very important in the context of relative motion, as the relative motion is calculated with respect to the reference frames. For some bigger reference frame, we can take the example of the motion of earth moving around the sun, in which solar system is the reference frame with respect to the earth moving around the sun. To calculate the relative motion of the objects, we first have to calculate the values of relative velocity and relative acceleration with respect to the reference frame.

Relative velocity:

Relative velocity should be taken into consideration to calculate the value of the relative motion of any object like an airplane in the wind or the boat in the stream. The acceleration, velocity, and position are very important in the process of calculation of relative motion depending on its reference frame. The relative velocity of any object can be defined as when two objects are in the same motion, and the relative velocity of the object A with respect to the relative velocity of object B is called as the relative velocity of that object. The vector sum of relative velocity is included in the formulae of relative velocity, which can be shown mathematically as;

vPE = vPT + vTE

The above equation of the relative velocity can be described as the velocity of P with respect to E is equal to the velocity of P with respect to the velocity of T plus the velocity of T with respect to the velocity of E, which gives the overall relative velocity of that object. The reference frame in this above equation is T which is the intermediate reference frame of the object. This approach of the relative velocity can be taken into consideration in many other examples like an airplane or boat, etc. The general formulae of relative velocity can be given as:

Velocity of P is relative to E = vP − vE

When the two objects are moving in the same direction at the same relative motion in which the relative velocity of P is relative to the relative velocity of E. But if the two objects are moving in the opposite directions with the same relative motion in which the relative velocity of E is relative to the relative velocity of P. The main difference between the moving directions of the objects determines the sign between the velocities of the two objects, which can be shown mathematically as,
vPE = vP + vE (when the two objects are moving in the same directions, the sign between the velocities of two objects are positive).

vEP = vE − vP (when the two objects are moving in the opposite direction, the sign between the velocities of two objects are negative).

Relative Acceleration:

The relative acceleration can be defined as the comparison of the acceleration of two different objects or rigid bodies moving in the same direction or the opposite direction. Basically, acceleration is the rate of change of velocity at a particular time interval. The relative acceleration can be determined by its general formulae given as follows;
aA = aB + aA B

The above general form of the relative acceleration gives the acceleration of point A which is equal to the acceleration of point B with the acceleration of point A with respect to the acceleration of point B.

With the help of the above information of relative velocity and relative acceleration, we can easily calculate the value of relative motion. Relative motion velocity can further be explained with the help of other dimensional examples shown below;

Relative motion velocity in one dimension:

There are only two possible ways for the movement of objects which are either in the same direction or the opposite direction. In this, one dimensional way is used in which the objects are moving with the positive velocity to form relative motion. For e.g., if a person is sitting inside a moving train, which is moving in the positive direction, with respect to the reference frame (earth is taken as a reference frame in this type of examples). In this type of examples, the value of velocity with respect to the earth can be written as vTE = 10 m/s, where TE refers to the train. But if the person sitting inside the train starts moving backward having the speed of 2 m/s, which is also known as the relative velocity with respect to the reference frame train. Since the person is moving in the opposite direction so the relative velocity of the person with respect to the reference frame of the train is vPT = -2 m/s, where vPT is the relative velocity of the person moving in opposite direction inside the train. The above two relative velocity factors can be added to form the total relative velocity of the train and the person with respect to the reference frame of earth, i.e.

VPE = VPT + VTE

So, through the above equation we can easily find the total relative velocity ( VPE ) of the train and the person with respect to the reference frame of earth.

Relative motion velocity in two dimensions:

In this two dimensional way of considering the relative motion velocity, two objects are taken A and B which are moving with the velocity of VA and VB in opposite direction, due to which the negative sign is taken in the value of relative velocity, with respect to the reference frame ( ground or earth ). There can be two conditions in this type of velocity calculation, which are the velocity of object A with respect to the velocity of object B and the velocity of the object A with respect to the velocity of the object B, mathematically it can be stated as below;

Vab = Va – Vb (where, va is the velocity of object A, vb is the velocity of object B and the velocity of object A with respect to the velocity of object B is given by vab).

Similarly, if the velocity of object B is taken with respect to the velocity of the object B can be expressed mathematically by the given equation;

Vba = Vb – Va (where, vb is the velocity of object B, va is the velocity of object A and the velocity of object B with the reference of object A is indicated as Vba).

If the velocity of object A (Va) and the velocity of object B (Vb) are of the same sign (moving in the same direction), then the magnitude of Vab and Vba will be lower than the magnitude of Vb and Va. But if the velocity of object A (Va) and the velocity of object B (Vb) are of the different sign (moving in the opposite direction), then the magnitude of Vab and Vba will be higher than the magnitude of Vb and Va.

There is a special case of measuring relative motion velocity, in which both the objects taken with the reference frame are kept stationary with each other, due to which the relative velocity of both the objects are taken 0, i.e.
Vb = Va
Vba = Vab = 0

Some examples of relative motion velocity with respect to the reference frame can be shown as follows:-

  • • Motion of a car relative to the truck (where the car and the person driving the car are considered as two objects and the truck is taken as a reference frame)

  • • Flying a plane in wind (where the plane and the person flying the plane are considered as two objects and the wind is taken as a reference frame)

  • • A man rowing a boat in the river (where the boat and man rowing the boat are considered as two objects and the river is taken as a reference frame).