5.8 Relative motion of projectiles  (Page 2/4)

The relative velocity in x-direction is :

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

As horizontal component of velocity of projectile does not change with time, we can re-write the equation of component relative velocity as :

Significantly, relative velocity in x-direction is determined by the initial velocities or initial conditions of the projection of two projectiles. It is expected also as components of velocities in x-direction do not change with time. The initial velocity, on the other hand, is fixed for a given projectile motion. As such, horizontal component of relative velocity is constant. A plot of relative velocity in x - direction .vs. time will be a straight line parallel to time axis.

The separation between two objects in x-direction at a given time "t" depends on two factors : (i) the initial separation of two objects in x-direction and (ii) relative velocity in x - direction. The separation in x - direction is given as :

$$\Delta x=x_A-x_B=x_0+v_{ABx}t$$

where $x_0$ is the initial separation between two projectiles in x-direction. Clearly, the separation in horizontal direction .vs. time plot would be a straight line with a constant slope. In physical terms, the separation between two projectiles in horizontal direction keeps increasing at a constant rate, which is equal to the magnitude of the component of relative velocity in that direction.

Relative velocity in vertical direction

The motion in the vertical direction, however, is subject to acceleration due to gravity, which always acts in vertically downward direction. The relative velocity in y-direction is :

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

As vertical component of motion is not a uniform motion, we can use equation of motion to determine velocity at a given time “t” as,

$$v_{Ay}=u_{Ay}-gt$$

$$v_{By}=u_{By}-gt$$

Putting in the expression of relative velocity in y-direction, we have :

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

The important aspect of the relative velocity in vertical y-direction is that acceleration due to gravity has not made any difference. The component relative velocity in y-direction is equal to simple difference of components of initial velocities of two projectiles in vertical direction. It is clearly due to the fact the acceleration of two projectiles in y-direction are same i.e. acceleration due to gravity and hence relative acceleration between two projectiles in vertical direction is zero. It means that the nature of relative velocity in vertical direction is same as that in the horizontal direction. A plot of relative velocity in y -direction .vs. time will be a straight line parallel to time axis.

The separation between two objects in y-direction at a given time "t" depends on two factors : (i) the initial separation of two objects in y-direction and (ii) relative velocity in y - direction. The separation in y - direction is given as :

$$\Delta y=y_A-y_B=y_0+v_{ABy}t$$

where $y_0$ is the initial separation between two projectiles in y-direction. Note that acceleration term has not appeared in the expression of relative velocity, because they cancel out. Clearly, the separation in vertical direction .vs. time plot would be a straight line with a constant slope. In physical terms, the separation between two projectiles in vertical direction keeps increasing at a constant rate, which is equal to the magnitude of the component of relative velocity in that direction.