Until now, we have discussed those cases when the frame of reference is xed, but in certain applications, it is often advantages to analyse the motion wrt moving frame of reference. This approach is termed as relative-motion analysis. Consider a frame of reference S’ which is moving wrt a xed frame of reference S as shown in the figure
Let us assume a particle P, whose position wrt S is described by
at any instant while
are the position of P wrt S’ and of S’ wrt S respectively at the same instant, then from vector addition,
Differentiating above equation wrt time we get,
Velocity of P wrt S = velocity of P wrt S’ + velocity of S’ wrt S
If we differentiate the last equation wrt time then we get,
where symbols have their usual meanings.
The Eqs. (i), (ii), (iii) together are termed as equations of relative motion.
Consider two bodies A and B moving with velocity
respectively, then the velocity of A wrt B is,
important to note here that velocities of both the particles are wrt the same frame of reference.
From standard relative motion equation,
where G stands for ground the default frame of reference. As
The more meaningful equation.
Relative motions equation is vector equations.
♦The choice of the xed frame of reference S depends upon the nature of the problem. Like for most earth-bound engineering problems, it is sufciently precise to take a xed engineering problems, it is sufciently precise to take a xed frame of reference attached to the earth, in which we can neglect the motion of the earth, while if we want to analyse the motion of the satellites around the earth, a non-rotating frame of reference is chosen with its origin on the axis of rotation of the earth.
♦The equations of motion (corresponding to constant acceleration) can also be applied to the elative motion provided relative acceleration is constant.
If we drop a particle from a moving car as shown,
Let us say at the instant of dropping the ball the velocity of the car is v and acceleration of the car are constant say a. At the instant of dropping velocity of particle wrt car is 0 but wrt ground it has some velocity due to the motion of the car. From
While acceleration of particle wrt ground is remaining g only because only gravity effect of the earth is causing the particle to move.