Relative Motion Physics Notes

Relative Motion:
The relative velocity of the first body with respect to another body, when both are in motion, is the rate at which the first body changes its position with respect to another body.

The concept of relative motion or relative velocity is all about understanding the frame of reference. A frame of reference can be thought of as the state of motion of the observer of some event. For example, if you are sitting on a lawn chair watching train travel past you from left to right at 50 m/s. You would consider yourself in a stationary frame of reference from your perspective, you are at rest, and the train is moving. Further, assuming you have tremendous eyesight, you could even watch a glass of water on a table inside the train move from left to right at 50 m/s.

An observer on the train itself, however, sitting beside the table with a glass of water, would view the glass of water as remaining stationary from their frame of reference, because that the observer is moving at 50 m/s, and the glass of water is moving at 50 m/s, the observer on the train sees no motion for the glass of water.
Due to different frame of reference and their positions, the motion of an object relative to different reference frames can be similar or different.
Relative Motion Physics Notes 1
Position of a point in a different frame of references

According to the figure, let us discuss the two reference frames S and S’ whose origins O and O’ are relative and parallel. In the reference frame S,r is the vector of point P relative to O and in the reference frame S’, the vector is r0 with relative to O’. Then, in the reference frame S’, the vector r’ of point P relative to O’ will be given as;
r’ = r – r₀ …(1)
Differentiating the above equation with respect to t,
\(\frac{d r^{\prime}}{d t}=\frac{d r}{d t}-\frac{d r_{0}}{d t}\)
or v’= v – v₀ …(2)
Here, v0 is the velocity in the reference frame S’ relative to S.

Equation (2) depicts the relationship between the velocity vectors of the references S and S’.
Again differentiating equation (2) with respect to t.
\(\frac{d v^{\prime}}{d t}=\frac{d v}{d t}-\frac{d v_{0}}{d t}\)
or a’ = a – a₀ …(3)
Here, a0 is the acceleration of reference frame S’ relative to S.

If a₀ = 0 i. e., if frame S and S’ are moving relative to one another with definite velocity then; a’= a

Hence, in this state, the acceleration of any particle would be the same in both the frame of reference.

Considering in one-dimensional motion; if two objects A and B are moving in the direction of the x-axis with definite velocity v_A and v_B. At any instant the position of the objects would be x_A and x_B respectively, then the position of B relative to A would be;
x_BA = x_B – x_A ……..(4)
and B’s velocity relative to A would be;
v_BA = v_B – v_A ……….(5)

Note: Relative velocity is used to describe the motion of airplanes in the wind or moving boats through water etc. This velocity is computed according to the observer inside the object. This can be computed by introducing an intermediate frame of reference, in simpler words, this can be the vector sum of the velocities.

For example, the velocity of object A with respect to reference frame C would be written as V_AC. Even if you don’t know the velocity of object A with respect to C directly, by finding out the velocity of object A with respect to some intermediate object B, and the velocity of object B with respect to C, you can combine your velocities using vector addition to obtain:

\(\overrightarrow{v_{A C}}=\overrightarrow{v_{A B}}+\overrightarrow{v_{B C}}\)
This sounds more complicated than it actually is. Let’s look at how this is applied in few examples: