Relative Motion JEE Physics

Any two objects can be compared based on their different characteristics as relative. It is possible to define relative motion as comparing one object's motion with that of another object moving with the same velocity. Using relative velocity, relative acceleration, or relative speed, one can determine relative motion easily. As long as the relative motion between an object and the earth remains constant while the relative motion between the object and the earth remains constant, the object is in a static state. The concept of relative motion refers to the comparison of two rigid bodies' relative accelerations and velocities. It is easy to say that the motion of any two objects can be compared as relative motion.

The relative motion of an object with respect to another object is known as its reference frame. It is possible to calculate the relative motion of an object, such as a train if that object is compared to the earth. Thus, in this example, a train is travelling at a certain relative speed relative to Earth (which is our reference frame). When it comes to relative motion, reference frames matter greatly since relative motion is calculated according to reference frames. We can take the motion of the earth around the sun as an example of a bigger frame of reference, where the solar system is the reference frame when it comes to the earth moving around the sun. Initially, we must determine the values of relative velocity and relative acceleration with respect to the reference frame in order to calculate the relative motion of the objects.

Relative Velocity

When calculating the relative motion of any object, like a boat in the stream or an aeroplane in the wind, the relative velocity should be taken into account. A reference frame plays an important role in the calculation of relative motion based on acceleration, velocity, and position. Whenever two objects are moving in the same direction, the relative velocity of objects A and B can be defined as that object's relative velocity. Relative velocity formula include the vector sum of relative velocity, which can be expressed as follows:

vPE = vPT + vTE 

According to the above equation, relative velocity can be calculated as the difference between the velocity of P with respect to E and the difference between T and E with respect to P. The result is the overall relative velocity of the object. It is the object's intermediate reference frame, T, that is used in this equation. There are many other examples where relative velocity can be considered, such as an aeroplane or boat. There are several formulas that describe relative velocity:

The velocity of P is relative to E = vP − vE 

It is possible to move the two objects in the same direction at the same relative motion where the relative velocity of P is relative to the relative velocity of E. If the two objects are moving in opposite directions together, with the relative velocity of P being greater than the relative velocity of E. It is this difference in the direction that determines the sign between the two velocities, which is shown mathematically as,

 vPE = vP + vE (the sign of the difference between two objects' velocity is positive when they are moving in the same direction).

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

Relative Acceleration

Relative acceleration is the difference between the accelerations of two different objects or rigid bodies moving in the same direction or in the opposite direction. The rate at which a velocity changes in a particular time period is acceleration. It can be calculated by using its general formulae, which are as follows;

aA = aB + aA B 

Using the above general form of relative acceleration, point A's acceleration is the same as the acceleration of point B's acceleration, relative to point B's acceleration.

We can easily calculate the value of relative motion based on the relative velocity and relative acceleration above. Below are some additional dimensional examples that help explain relative motion velocity;

The Velocity of Relative Motion in One Dimension: 

Objects can only move in one of two directions: either in the same direction or in the opposite direction. The objects are moving in a one-dimensional manner, with positive velocity, to produce relative motion. An example of this is if a person is sitting inside a moving train and the train is moving in the direction of the reference frame (the earth is assumed in this type of example). With this type of example, the value of velocity with respect to the earth can be written as vTE = 10 m/s, where TE stands for the train. The relative velocity is the speed at which the train is moving relative to a stationary person inside the train. Due to the direction of motion of the person, 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 moving person inside the train. With respect to the reference frame of earth, the above two relative velocity factors can be added together to determine the total relative velocity.

 VPE = VPT + VTE 

Through the equation above, we can easily determine the total relative velocity of the train and the individual relative to the reference frame of earth.

Relative Motion Velocity In Two Dimensions

The relative motion velocity is viewed in the two dimensions of A and B, which are moving with the velocity of VA and VB in opposing directions, therefore taking the negative sign of the value of relative velocity, as measured in reference to the reference frame (earth). During this type of velocity calculation, two conditions can be taken into consideration, namely the velocity of object A in relation to the velocity of object B, and the velocity of object A with respect to the velocity of object B, mathematically stated as follows;

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

Additionally, if the velocity of object B is taken with respect to the velocity of object B, then the following equation can be applied;

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

The magnitude of Vab and Vba will be lower than the magnitudes of Vb and Va if the velocity of A (Va) and B (Vb) is of the same sign (moving in the same direction). The magnitude of Vab and Vba will be larger than the magnitude of Vb and Va, however, if the velocity of object A (Va) and the velocity of object B (Vb) are of the opposite sign (moving in the opposite direction).

The measurement of relative motion velocity can be performed in a special case where both objects of interest are stabilised with respect to each other, thereby causing both of the objects to have 0 relative velocities, i.e.

Vb = Va 

Vba = Vab = 0