5.8 Relative motion of projectiles  (Page 3/4)

Resultant relative motion

The component relative velocities in two mutually perpendicular directions have been derived in the previous section as :

$$v_{ABx}=u_{Ax}-u_{Bx}$$

$$v_{ABy}=u_{Ay}-u_{By}$$

These equations are very important results. It means that relative velocity between projectiles is exclusively determined by initial velocities of the two projectiles i.e. by the initial conditions of the two projectiles as shown in the figure below. The component relative velocities do not depend on the subsequent motion i.e. velocities. The resultant relative velocity is vector sum of component relative velocities :

$${\mathbf{v}}_{AB}=v_{ABx}\mathbf{i}+v_{ABy}\mathbf{j}$$

Since the component relative velocities do not depend on the subsequent motion, the resultant relative velocity also does not depend on the subsequent motion. The magnitude of resultant relative velocity is given by :

The slope of the relative velocity of “A” with respect to “B” from x-direction is given as :

Physical interpretation of relative velocity of projectiles

The physical interpretation of the results obtained in the previous section will help us to understand relative motion between two projectiles. We recall that relative velocity can be interpreted by assuming that the reference object is stationary. Consider the expression, for example,

$$v_{ABx}=u_{Ax}-u_{Bx}$$

What it means that relative velocity “vABx” of object “A” with respect to object “B” in x-direction is the velocity of the object “A” in x-direction as seen by the stationary object “B”. This interpretation helps us in understanding the nature of relative velocity of projectiles.

Extending the reasoning, we can say that object “A” is moving with uniform motion in “x” and “y” directions as seen by the stationary object “B”. The resultant motion of “A”, therefore, is along a straight line with a constant slope. This result may be a bit surprising as we might have expected that two projectiles see (if they could) each other moving along some curve - not a straight line.