Question

The vector position of a 3.05 g particle moving in the xy plane varies in time according to r with arrow1 = 3i + 3j t + 2jt2 where t is in seconds and r with arrow is in centimeters. At the same time, the vector position of a 5.15 g particle varies as r with arrow2 = 3i − 2it2 − 6jt.

Answer 1

masses m = 3.05 g

            m ' = 5.15 g

Position vectors r = 3i+3j t +2j t ²

                       r ' = 3i − 2it² − 6jt

at time t= 2.6 , r = 3i+3j t +2j t ²

                         = 3 i + 3(2.6 )j +2(2.6) ² j

                         = 3 i + 7.8 j + 13.52 j

                         = 3 i +21.32 j

                       r ' = 3i − 2it² − 6jt

                           = 3i-2(2.6) ² i -6(2.6) j

                           = 3 i - 13.52 i - 15.6 j

                           = -10.52 i - 15.6 j

Position of center of mass r cm = (mr +m ' r ' ) / ( m+ m' )

                                              = [3.05(3 i +21.32 j) +5.15(-10.52 i-15.6 j ) ]/[3.05+5.15]

                                              =[ 9.15 i +65.026 j -54.178 i - 80.34 j]/8.2

                                              = [ -45.028 i -15.314 j] / 8.2

                                              = -5.4912 i - 1.867 j

Velocity of the 1 st particle v = dr / dt

                                           = d(3i+3j t +2j t ²) / dt

                                           = 3 j + 4t j

Velocity of the 2 nd particle v ' = dr ' / dt

                                             = d( 3i − 2it² − 6jt ) / dt

                                             = -4t i -6 j

At time t= 2.6 s   v = 3 j + 4(2.6) j

                            = 3 j + 10.4 j

                           = 13.4 j

                        v ' = -4(2.6) i -6 j

                           = -10.4 i - 6 j

V cm = [mv + m' v ' ] /(m+ m' )

         = [3.05(13.4 j ) + 5.15 (-10.4 i -6 j) ] / [3.05 +5.15 ]

         = [40.87 j -53.56 i - 30.9 j ] 8.2

         = [-53.56 i + 9.97 j ]/8.2

         = -6.531 i +1.215 j