1. A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass \(M\), of which a mass \(k M , k < 1\), is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed \(c\), relative to the rocket, in a direction opposite to that of the rocket's motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt.
2. Two forces \(\mathbf { F } _ { 1 } = ( 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) act on a rigid body.

The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector ( \(2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) ) m and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(- 3 \mathbf { i } + 2 \mathbf { k }\) ) m.
The two forces are equivalent to a single force \(\mathbf { R }\) acting at the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple of moment \(\mathbf { G }\). Find,

1. \(\mathbf { R }\),
2. \(\mathbf { G }\). A third force \(\mathbf { F } _ { 3 }\) is now added to the system. The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector ( \(2 \mathbf { i } - \mathbf { k }\) ) m and the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a couple.
3. Find the magnitude of the couple.