3.2 Relative velocity in two dimensions  (Page 2/3)

$$\begin{array}{l}\frac{đ{\mathbf{r}}_{CA}}{đt}=\frac{đ{\mathbf{r}}_{BA}}{đt}+\frac{đ{\mathbf{r}}_{CB}}{đt}\end{array}$$

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

The meaning of the subscripted velocities are :

• ${\mathbf{v}}_{CA}$ : velocity of object "C" with respect to "A"
• ${\mathbf{v}}_{CB}$ : velocity of object "C" with respect to "B"
• ${\mathbf{v}}_{BA}$ : 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 :

$$\begin{array}{l}\Rightarrow \frac{đ{\mathbf{v}}_{CA}}{đt}=\frac{đ{\mathbf{v}}_{BA}}{đt}+\frac{đ{\mathbf{v}}_{CB}}{đt}\end{array}$$

$$\begin{array}{l}\Rightarrow {\mathbf{a}}_{CA}={\mathbf{a}}_{BA}+{\mathbf{a}}_{CB}\end{array}$$

The meaning of the subscripted accelerations are :

• ${\mathbf{a}}_{CA}$ : acceleration of object "C" with respect to "A"
• ${\mathbf{a}}_{CB}$ : acceleration of object "C" with respect to "B"
• ${\mathbf{a}}_{BA}$ : 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. ${\mathbf{a}}_{BA}=0$ . Hence,

$$\begin{array}{l}\Rightarrow {\mathbf{a}}_{CA}={\mathbf{a}}_{CB}\end{array}$$

The observers moving at constant velocities, therefore, measure same acceleration of the object (C).

Interpretation of the equation of relative velocity

The interpretation of the equation of relative motion in two dimensional motion is slightly tricky. The trick is entirely about the ability to analyze vector quantities as against scalar quantities. There are few alternatives at our disposal about the way we handle the vector equation.

In broad terms, we can either use graphical techniques or vector algebraic techniques. In graphical method, we can analyze the equation with graphical representation along with analytical tools like Pythagoras or Parallelogram theorem of vector addition as the case may be. Alternatively, we can use vector algebra based on components of vectors.

Our approach shall largely be determined by the nature of inputs available for interpreting the equation.

Equation with reference to earth

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

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

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.