Relative velocity in one dimension  (Page 4/4)

This technique is a very useful tool for consideration of relative motion in two dimensions.

Direction of relative velocities

For a pair of two moving objects moving uniformly, there are two values of relative velocity corresponding to two reference frames. The values differ only in sign – not in magnitude. This is clear from the example here.

Relative velocity .vs. difference in velocities

It is very important to understand that relative velocity refers to two moving bodies – not a single body. Also that relative velocity is a different concept than the concept of "difference of two velocities", which may pertain to the same or different objects. The difference in velocities represents difference of “final” velocity and “initial” velocity and is independent of any order of subscript. In the case of relative velocity, the order of subscripts are important. The expression for two concepts viz relative velocity and difference in velocities may look similar, but they are different concepts.

Relative acceleration

We had restricted out discussion up to this point for objects, which moved with constant velocity. The question, now, is whether we can extend the concept of relative velocity to acceleration as well. The answer is yes. We can attach similar meaning to most of the quantities - scalar and vector both. It all depends on attaching physical meaning to the relative concept with respect to a particular quantity. For example, we measure potential energy (a scalar quantity) with respect to an assumed datum.

Extending concept of relative velocity to acceleration is done with the restriction that measurements of individual accelerations are made from the same reference.

If two objects are moving with different accelerations in one dimension, then the relative acceleration is equal to the net acceleration following the same working relation as that for relative velocity. For example, let us consider than an object designated as "1" moves with acceleration " $a_1$ " and the other object designated as "2" moves with acceleration " $a_2$ " along a straight line. Then, relative acceleration of "1" with respect to "2" is given by :

$$\begin{array}{l}\Rightarrow a_{12}=a_1-a_2\end{array}$$

Similarly,relative acceleration of "2" with respect to "1" is given by :

$$\begin{array}{l}\Rightarrow a_{21}=a_2-a_1\end{array}$$

Worked out problems

Check your understanding

Check the module titled Relative velocity in one dimension (Check your understanding) to test your understanding of the topics covered in this module.