The problem can be solved using the principle of impulse and momentur since it involves fore, velocity, and time, First draw the FBD.

Ex.5.4 A particle moves such that its velocity varies according to the relation v =v,- kx where k is a constant and v, is the velocity in the position P where x = 0 and t = 0. Derive the expressions for the position coordinate x, velocity v and acceleration a in terms of the time t.

Pr5. A yo-yo is a toy consisting of an axle of radius r connected to two disks of radius R, and a thread looped around the axle. The end of the thread is fixed to the ceiling and the yo-yo rolls down so that the axis moves along a vertical line. (The thread does not slip on the axle.) R. a) Find the velocity of point P indicated in the figure in the moment when the axis of the yo-yo moves with velocity v. Express your answer in terms of r, R and v. b) Find the radius of curvature of the trajectory of point P in the moment shown in the figure. Give your answer in terms of r and R.