The vertical motion of mass Ais defined by the relation 10 sin 215cos 2100,xtt=++where xand tare expressed in mm and seconds, respectively. Determine (a) the position, velocity and acceleration of Awhen t=1 s, (b) the maximum velocity and acceleration of A.

Solution:

               x = 10sin 2t + 15cos 2t +100

               v = dx/dy = 20cos 2t - 30sin 2t

               a = dv/dt = 40sin 2t - 60cos 2t

For trigonometric function set calculator to radian:

(a) At t = 1 s                 x1 = 10sin 2 + 15cos 2 + 100 = 102.9

                                     v1 = 20cos 2 - 30sin 2 = -35.6

                                     a1 = -40sin 2 - 60cos 2 = -11.40

(b)  Maximum velocity occure when a = 0.

                                     -40sin 2t - 60cos 2t = 0

                                       tan 2t = -60/40 = -1.5

                                       2t = tan⁻¹ (-1.5) = -0.9828  and -0.9828 +

Reject the negative value.   2t = 2.1588

                                                 t = 1.0794

                                                 t = 1.0794  for V

   so                                   =20cos(2.1588) -30sin(2.1588)

                                                    = -36.056

Note that we could have also used

                                                          = 20² +30²

                                                                     = 36.056

by combining the sine and cosine terms.

For  we can take the derivative and set equal to zero or just combine the sine and cosine terms.

                                    =  

                                              = 72.1 mm/