Relative velocity in one dimension  (Page 2/4)

$$\begin{array}{l}{v}_{CA}=\frac{đx_{CA}}{đt}\end{array}$$

and

$$\begin{array}{l}{v}_{CB}=\frac{đx_{CB}}{đt}\end{array}$$

Now, we can obtain relation between these two velocities, using the relation $x_{CA}=x_{BA}+x_{CB}$ and differentiating the terms of the equation with respect to time :

$$\begin{array}{l}\frac{đx_{CA}}{đt}=\frac{đx_{BA}}{đt}+\frac{đx_{CB}}{đt}\end{array}$$

$$\begin{array}{l}\Rightarrow v_{CA}=v_{BA}+v_{CB}\end{array}$$

The meaning of the subscripted velocities are :

• $v_{CA}$ : velocity of object "C" with respect to "A"
• $v_{CB}$ : velocity of object "C" with respect to "B"
• $v_{BA}$ : velocity of object "B" with respect to "A"

Acceleration of the point object

If the object being observed is accelerated, then its acceleration is obtained by the time derivative of velocity. Differentiating equation of relative velocity, we have :

$$\begin{array}{l}{v}_{CA}=v_{BA}+v_{CB}\end{array}$$

$$\begin{array}{l}\Rightarrow \frac{đv_{CA}}{đt}=\frac{đv_{BA}}{đt}+\frac{đv_{CB}}{đt}\end{array}$$

$$\begin{array}{l}\Rightarrow a_{CA}=a_{BA}+a_{CB}\end{array}$$

The meaning of the subscripted accelerations are :

• $a_{CA}$ : acceleration of object "C" with respect to "A"
• $a_{CB}$ : acceleration of object "C" with respect to "B"
• $a_{BA}$ : acceleration of object "B" with respect to "A"

But we have restricted ourselves to reference systems which are moving at constant velocity. This means that relative velocity of "B" with respect to "A" is a constant. In other words, the acceleration of "B" with respect to "A" is zero i.e. $a_{BA}=0$ . Hence,

$$\begin{array}{l}\Rightarrow a_{CA}=a_{CB}\end{array}$$

The observers moving at constant velocity, therefore, measure same acceleration of the object. As a matter of fact, this result is characteristics of inertial frame of reference. The reference frames, which measure same acceleration of an object, are inertial frames of reference.

Interpretation of the equation of relative velocity

The important aspect of relative velocity in one dimension is that velocity has only two possible directions. We need not use vector notation to write or evaluate equation of relative velocities in one dimension. The velocity, therefore, can be treated as signed scalar variable; plus sign indicating velocity in the reference direction and minus sign indicating velocity in opposite to the reference direction.

Equation with reference to earth

The equation of relative velocities refers velocities in relation to different reference system.

$$\begin{array}{l}{v}_{CA}=v_{BA}+v_{CB}\end{array}$$

We note that two of the velocities are referred to A. In case, “A” denotes Earth’s reference, then we can conveniently drop the reference. A velocity without reference to any frame shall then mean Earth’s frame of reference.