A particle is initially at rest at the point which is metres to the right of

The particle then moves in a straight line towards

For the acceleration of the particle is given by    where is the distance from and is a positive constant.

1. Prove that    (2 marks)
   

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2. Using the substitution   show that the time taken to reach a distance metres to the right of is given by
    
          (3 marks)
   

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It can be shown that    (Do NOT prove this.)

1. What is the limiting time taken for the particle to reach   (1 mark)
   

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In an alien universe, the gravitational attraction between two bodies is proportional to  , where    is the distance between their centres.

A particle is projected upward from the surface of a planet with velocity    at time  .  Its distance    from the centre of the planet satisfies the equation

1. Show that  , where    is the magnitude of the acceleration due to gravity at the surface of the planet and    is the radius of the planet.  (1 mark)
   

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2. Show that  , the velocity of the particle, is given by
    
          (3 marks)
   

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3. It can be shown that   (Do NOT prove this.)
   

    

   Show that if    the particle will not return to the planet.  (2 marks)
   

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4. If    the particle reaches a point whose distance from the centre of the planet is  , and then falls back.

5.  

     (1)   Use the formula in part (ii) to find    in terms of    and    (1 mark)
   

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6.  

     (2)   Use the formula in part (iii) to find the time taken for the particle to return to the surface of the planet in terms of    and    (1 mark)
   

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