8 3. A particle P of mass 0.5 kg is moving along the positive x-axis in the direction of x increasing. At time t seconds (t  0) , P is x metres from the origin O and the speed of P is v m s−1. The resultant force acting on P is directed towards O and has magnitude kv2 N, where k is a positive constant. When x = 1 , v = 4 and when x = 2 , v = 2 (a) Show that v = ab x , where a and b are constants to be found. (6) The time taken for the speed of P to decrease from 4 m s−1 to 2 m s−1 is T seconds. (b) Show that T = 1 4 2 ln (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________