The maximum speed that a hammer can reach after falling vertically from the top of a skyscraper is .

The hammer's  speed, , after falling metres, is given by the differential equation 

  where is a positive constant.

Find an expression of in terms of .    (4 marks)

The velocity of a particle satisfies the differential equation  ,  where    centimetres is its displacement relative to a fixed point at time seconds.

Initially, the displacement of the particle is 1 cm.

1. Find an expression for in terms of .  (3 marks)
   

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2. Find the maximum displacement of the particle and the times at which this occurs.  (2 marks)
   

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