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Mechanics, EXT2 M1 2019 HSC 14b

A parachutist jumps from a plane, falls freely for a short time and then opens the parachute. Let t be the time in seconds after the parachute opens,   be the distance in metres travelled after the parachute opens, and    be the velocity of the parachutist in .

The acceleration of the parachutist after the parachute opens is given by

where is the acceleration due to gravity and is a positive constant.

  1. With an open parachute the parachutist has a terminal velocity of  .

     

    Show that  .  (1 mark)

    At the time the parachute opens, the speed of descent is .

  2. Show that it takes    seconds to slow down to a speed of .  (4 marks)

  3. Let    be the distance the parachutist travels between opening the parachute and reaching the speed .

     

     

    Show that  .  (3 marks)

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i.   

 

ii.   

 

 

 

 

 
 
 

 

iii.   
 
 
 
   
   

 

 
 
 
 
 
 

Mechanics, EXT2 M1 2011 HSC 6a

Jac jumps out of an aeroplane and falls vertically. His velocity at time    after his parachute is opened is given by  , where    and    is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is  , where    is a positive constant. Let    be Jac’s mass and    the acceleration due to gravity. Jac’s terminal velocity with the parachute open is  

Jac’s equation of motion with the parachute open is

   (Do NOT prove this.)

  1. Explain why Jac’s terminal velocity    is given by  
     
           (1 mark)

  2. By integrating the equation of motion, show that    and    are related by the equation
     
           (3 marks)

  3. Jac’s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes.

     

    Jac opens his parachute when his speed is   Gil opens her parachute when her speed is   Jac’s speed increases and Gil’s speed decreases, both towards  

     

    Show that in the time taken for Jac's speed to double, Gil's speed has halved.  (3 marks)

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i.  

  

ii. 

  ­

  

♦ Mean mark part (ii) 39%.

­
 
 
 
 
 

 

 

iii. 

 
 
 

 

 

♦♦♦ Mean mark part (iii) 16%.