Teaching Relative Motion
The position, velocity and acceleration of a particle are relative terms and are defined with respect to an assumed reference point. The reference point may be a t rest (that is, stationary on the ground), moving with constant velocity or uniform acceleration. So far, we have studied the motion of a particle with respect to a stationary point. In this section we will modify the equations of kinematics with respect to a moving point.
Suppose two objects A and B are in motion with respect to a fixed origin O as well as with respect to each other. Their position vectors at any time t are rA/O and rB/O respectively and the vector joining B with A is rB/A. By using vector diagram
rB/O = rB/A + rA/O (1)
On differentiating the above equation with respect to time, we get
d/dt rB/O = d/dt rB/A + d/dt rA/O
VB/O = VB/A + VA/O
Velocity of B = Velocity of B + Velocity of A
w.r.t O w.r.t. A w.r.t O
The above relation is so obvious that we could have written it without intermediate steps. In the above discussion we have used double subscripts for velocity because by doing so we can write the final equation quite easily this is because the double subscript form equality under multiplication i.e.
B/O = (B/A)(A/O)
Further, we can extend this concept when number of objects are more than two. For example
VC/D = VC/B + VB/A + VA/O
If we further differentiate equation with respect to time, we get following equation for acceleration
aB/O = aB/A + aA/O