6 A particle \(P\) of mass 3 kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf { F } \mathrm { N }\). At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P , \mathbf { v } \mathrm {~ms} ^ { - 1 }\), is given by $$\mathbf { v } = ( 32 \sinh ( 2 t ) ) \mathbf { i } + ( 32 \cosh ( 2 t ) - 257 ) \mathbf { j } .$$

1. 1. By considering kinetic energy, determine the work done by \(\mathbf { F }\) over the interval \(0 \leqslant t \leqslant \ln 2\).
   2. Explain the significance of the sign of the answer to part (a)(i).
2. Determine the rate at which \(\mathbf { F }\) is working at the instant when \(P\) is moving parallel to the i-direction.