A particle of mass 1 kg is projected from the origin with speed 40 m s at an angle 30° to the horizontal plane.

1. Use the information above to show that the initial velocity of the particle is
2.     .   (1 mark)
   

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The forces acting on the particle are gravity and air resistance. The air resistance is proportional to the velocity vector with a constant of proportionality 4 . Let the acceleration due to gravity be 10 m s .

The position vector of the particle, at time seconds after the particle is projected, is and the velocity vector is .
 

2. Show that    (3 marks)
   

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3. Show that    (2 marks)
   

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4. The graphs    and  are given in the diagram below.
    
    
5. Using the diagram, find the horizontal range of the particle, giving your answer rounded to one decimal place.  (2 marks)
   

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