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Mechanics, EXT2 M1 2020 HSC 12b

A particle is projected from the origin with initial velocity m/s at an angle    to the horizontal. The particle lands at    on the -axis. The acceleration vector is given by  , where is the acceleration due to gravity. (Do NOT prove this.)
 

  1. Show that the position vector    of the particle is given by
     
         .   (3 marks)

  2. Show that the Cartesian equation of the path of flights is given by
     
         .   (3 marks)

  3. Given  , prove that there are 2 distinct values of    for which the particle will land at  .   (2 marks)

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i.   


 

 


 


 


 

ii.   


 

 
 
 
 

 

iii.   

Mean mark part (iii) 52%.
   
 
  
 

 

 
 
 
 
 
 

 

Mechanics, EXT2* M1 2007 HSC 7b

A small paintball is fired from the origin with initial velocity metres per second towards an eight-metre high barrier. The origin is at ground level, metres from the base of the barrier.

The equations of motion are

where    is the angle to the horizontal at which the paintball is fired and    is the time in seconds. (Do NOT prove these equations of motion)
 

  1. Show that the equation of trajectory of the paintball is
     
         , where  .  (2 mark)

  2. Show that the paintball hits the barrier at height    metres when
     
         .

     

    Hence determine the maximum value of  .  (2 marks)

  3. There is a large hole in the barrier. The bottom of the hole is metres above the ground and the top of the hole is metres above the ground. The paintball passes through the hole if    is in one of two intervals. One interval is  .

     

    Find the other interval.  (2 marks)

  4. Show that, if the paintball passes through the hole, the range is 
     
         
     
    Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier.  (3 marks)

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i.   
 

 

 
 
 
 

 

ii. 

 

 
 
 
 

 

 

 

iii.   EXT1 2007 7bi

 
 
 

 

 
 
 

 

 

iv. 

 

 

 

 

 

Mechanics, EXT2* M1 2004 HSC 6b

A fire hose is at ground level on a horizontal plane. Water is projected from the hose. The angle of projection, , is allowed to vary. The speed of the water as it leaves the hose, metres per second, remains constant. You may assume that if the origin is taken to be the point of projection, the path of the water is given by the parametric equations

where    is the acceleration due to gravity.  (Do NOT prove this.)

  1. Show that the water returns to ground level at a distance  metres from the point of projection.   (2 marks)

This fire hose is now aimed at a 20 metre high thin wall from a point of projection at ground level 40 metres from the base of the wall. It is known that when the angle    is 15°, the water just reaches the base of the wall.  
 

Calculus in the Physical World, EXT1 2004 HSC 6b
 

  1. Show that  .  (1 mark)

  2. Show that the cartesian equation of the path of the water is given by
     
         .  (2 marks)

  3. Show that the water just clears the top of the wall if
     
         .  (2 marks)

  4. Find all values of    for which the water hits the front of the wall.  (2 marks)

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i.   
 

 

 

 
 

 

 

ii.   

 

iii. 

 

 
 
 
 

 

iv.   

 

v.   

Calculus in the Physical World, EXT1 2004 HSC 6b Answer

 

 
 

 
 
       
       

 

 

Mechanics, EXT2* M1 2014 HSC 14a

The take-off point on a ski jump is located at the top of a downslope. The angle between the downslope and the horizontal is  .  A skier takes off from with velocity m s−1 at an angle to the horizontal, where  .  The skier lands on the downslope at some point , a distance metres from .
 

2014 14a
 

The flight path of the skier is given by

,      (Do NOT prove this.)

where    is the time in seconds after take-off.

  1. Show that the cartesian equation of the flight path of the skier is given by
     
         .   (2 marks)

  2. Show that  
     
         .   (3 marks)

  3. Show that  
     
         .   (2 marks)

  4. Show that has a maximum value and find the value of for which this occurs.   (3 marks)

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i. 

 
 

 

ii. 

♦♦ Mean mark 28%
MARKER’S COMMENT: The key to finding and solving for is to realise you need the intersection of the Cartesian equation in (i) and the line .

 

 

 

 

iii. 

 
 

 

iv. 

♦ Mean mark 47%
IMPORTANT: Note that students can attempt every part of this question, even if they couldn’t successfully prove earlier parts.