The direction of velocity v at a time t is along

The direction of velocity v at a time t is along the tangent to the circle at the point where the particle is located at that instant. From the geometry of Fig. . , it is clear that the velocity of the projection particle P ′ at time t is v ( t ) = – ω A sin ( ω t + φ ) ( . ) where the negative sign shows that v (t) has a direction opposite to the positive direction of x -axis. Eq. ( . ) gives the instantaneous velocity of a particle executing SHM, where displacement is given by Eq. ( . ). We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. . ) with respect of t : d ( ) d v(t) = x t ( . ) The method of reference circle can be similarly used for obtaining instantaneous acceleration of a particle undergoing SHM. We know that the centripetal acceleration of a particle P in uniform circular motion has a magnitude v /A or ω A, and it is directed towards the centre i.e., the direction is along PO. The instantaneous acceleration of the projection particle P ′ is then (See Fig. . ) a ( t ) = – ω A cos ( ω t + φ ) = – ω x ( t ) ( . ) Fig. . The velocity, v (t), of the particle P ′ is