3.02e Two-dimensional constant acceleration: with vectors

A girl can paddle her canoe at \(5 \text{ m s}^{-1}\) in still water. She wishes to cross a river which is \(100 \text{ m}\) wide and flowing at \(8 \text{ m s}^{-1}\).

1. 1. Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
   2. Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
2. Calculate the times taken for each of the two cases in part (i). [3]

A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.

1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]

The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.

1. Find the position vector of the particle \(3\) seconds later. [5]