5.8 Relative motion of projectiles

In this module, we shall apply the concept of relative velocity and relative acceleration to the projectile motion. The description here is essentially same as the analysis of relative motion in two dimensions, which was described earlier in the course except that there is emphasis on projectile motion. Besides, we shall extend the concept of relative motion to analyze the possibility of collision between projectiles.

We shall maintain the convention of subscript designation for relative quantities for the sake of continuity. The first letter of the subscript determines the “object”, whereas the second letter determines the “other object” with respect to which measurement is carried out. Some expansion of meaning is given here to quickly recapitulate uses of subscripted terms :

${\mathbf{v}}_{AB}$ : Relative velocity of object “A” with respect to object “B”

$v_{ABx}$ : Component of relative velocity of object “A” with respect to object “B” in x-direction

For two dimensional case, the relative velocity is denoted with bold type vector symbol. We shall , however, favor use of component scalar symbol with appropriate sign to represent velocity vector in two dimensions like in the component direction along the axes of the coordinate system. The generic expression for two dimensional relative velocity are :

In vector notation :

In component scalar form :

Relative velocity of projectiles

The relative velocity of projectiles can be found out, if we have the expressions of velocities of the two projectiles at a given time. Let “ ${\mathbf{v}}_A$ ” and “ ${\mathbf{v}}_B$ ” denote velocities of two projectiles respectively at a given instant “t”. Then :

$${\mathbf{v}}_A=v_{Ax}\mathbf{i}+v_{Ay}\mathbf{j}$$

$${\mathbf{v}}_B=v_{Bx}\mathbf{i}+v_{By}\mathbf{j}$$

Hence, relative velocity of projectile “A” with respect to projectile “B” is :

$${\mathbf{v}}_{AB}={\mathbf{v}}_A-{\mathbf{v}}_B=v_{Ax}\mathbf{i}+v_{Ay}\mathbf{j}-v_{Bx}\mathbf{i}-v_{By}\mathbf{j}$$

We can interpret this expression of relative velocity as equivalent to consideration of relative velocity in component directions. In the nutshell, it means that we can determine relative velocity in two mutually perpendicular directions and then combine them as vector sum to obtain the resultant relative velocity. Mathematically,

where,

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

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

This is a significant analysis simplification as study of relative motion in one dimension can be done with scalar representation with appropriate sign.

Interpretation of relative velocity of projectiles

The interpretation is best understood in terms of component relative motions. We consider motion in both horizontal and vertical directions.

Relative velocity in horizontal direction

The interpretation is best understood in terms of component relative motion. In horizontal direction, the motion is uniform for both projectiles. It follows then that relative velocity in horizontal x-direction is also a uniform velocity i.e. motion without acceleration.