The position of a moving body is given by  , where is measured in seconds, for  .

The motion of the body can be described as moving along a parabolic path given by

1. , starting at , reversing direction at and then again at , then returning to after seconds.
2. , starting at , reversing direction at , then returning to after seconds.
3. , starting at , reversing direction at and then again at , then returning to after seconds.
4. , starting at , reversing direction at , then returning to after seconds.

A particle is moving along the -axis with velocity  , where    is a real constant.

At time  , a force acts on the particle, causing it to accelerate with acceleration  , where    is a negative real constant.

Which one of the following statements correctly describes the motion of the particle for  ?

1. The particle show down, stops momentarily and then begins to move in the opposite direction to its original motion.
2. The particle continues to travel along the -axis with decreasing speed.
3. The particle travels parallel to the -axis.
4. The particle moves along a circular arc.
5. The particle moves along a parabola.

The acceleration vector of a particle that starts from rest is given by

, where .

The velocity vector of the particle, , is given by

The position vector of a particle moving along a curve at time is given by  .

Find the length of the path that the particle travels along the curve from    to  .  (4 marks)

The position vector of a particle moving along  a curve at time seconds is given by 

   ,

where distances are measured in metres.

The distance metres that the particle travels along the curve in three-quarters of a second is given by

Find and , where  .  (5 marks)