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Mechanics, EXT2 M1 2021 HSC 16b

A particle which is projected from the origin with initial speed ms-1 at an angle of to the positive -axis lands on the -axis, as shown in the diagram. The particle is subject to an acceleration due to gravity of ms-1. 
 

The position vector of the particle, , where is the time in seconds after the particle is projected, is given by

.     (Do NOT prove this.)

For some value(s) of    there will be two times during the time of flight when the particle’s position vector is perpendicular to its velocity vector.

Find the value(s) of    for which this occurs, justifying that both times occur during the time of flight.  (5 marks)

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♦♦ Mean mark 31%.
 
 

 

 

 

 
 

 


 


 

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 2019 HSC 13c

Two objects are projected from the same point on a horizontal surface. Object 1 is projected with an initial velocity of  directed at an angle of    to the horizontal. Object 2 is projected 2 seconds later.

The equations of motion of an object projected from the origin with initial velocity at an angle to the -axis are

,

where    is the time after the projection of the object. Do NOT prove these equations.

  1. Show that Object 1 will land at a distance  m from the point of projection.  (2 marks)

  2. The two objects hit the horizontal plane at the same place and time.

     

    Find the initial speed and the angle of projection of Object 2, giving your answer correct to 1 decimal place.  (3 marks)

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a.   

 
 

 

 

 

 

(ii)   

♦ Mean mark 42%.

 

 

 

 

 
 

Mechanics, EXT2* M1 2017 HSC 13c

A golfer hits a golf ball with initial speed at an angle to the horizontal. The golf ball is hit from one side of a lake and must have a horizontal range of 100 m or more to avoid landing in the lake.
 

     

Neglecting the effects of air resistance, the equations describing the motion of the ball are

,

where is the time in seconds after the ball is hit and is the acceleration due to gravity in . Do NOT prove these equations.

  1. Show that the horizontal range of the golf ball is
     
          metres.  (2 marks)

  2. Show that if    then the horizontal range of the ball is less than 100 m.  (1 mark)

It is now given that    and that the horizontal range of the ball is 100 m or more.

  1. Show that  .  (2 marks)

  2. Find the greatest height the ball can achieve.  (2 marks)

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i.   

 

 
 

 

ii.   

♦ Mean mark 44%.

 

 

iii.   

 

iv.   

♦♦ Mean mark 35%.
 

 

 
 

 

 
 
 
 

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.
 

 

 
 

 

Mechanics, EXT2* M1 2012 HSC 14b

A firework is fired from , on level ground, with velocity metres per second at an angle of inclination  . The equations of motion of the firework are

 and. (Do NOT prove this.) 

The firework explodes when it reaches its maximum height.
 

2012 14b
 

  1. Show that the firework explodes at a height of    metres.   (2 marks)

  2. Show that the firework explodes at a horizontal distance of    metres from  .    (1 mark)

  3. For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from , and at least 150 m above the ground.

     

    For what values of    will this occur?   (3 mark)

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i.   


 

 

 
 

 

ii. 

 
 

 

iii. 

♦♦ Mean mark 28%
MARKER’S COMMENT: Many students ignored the findings in earlier parts and tried unsuccessfully to solve inequalities that still contained .

 

 

 

 

 

 

 

.

Mechanics, EXT2* M1 2009 HSC 6a

Two points, and ,  are on cliff tops on either side of a deep valley. Let and be the vertical and horizontal distances between and as shown in the diagram. The angle of elevation of from is  ,  so that  .
 

2009 6a
 

At time ,  projectiles are fired simultaneously from and .  The projectile from is aimed at , and has initial speed at an angle of    above the horizontal. The projectile from is aimed at and has initial speed at an angle    below the horizontal.

The equations of motion for the projectile from are

  and   ,

and the equations for the motion of the projectile from are

  and   ,     (DO NOT prove these equations.)

  1. Let be the time at which  .
     
    Show that  .   (1 mark)

  2. Show that the projectiles collide.    (2 marks)

  3. If the projectiles collide on the line  ,  where  ,  show that
     
         .    (1 mark)

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  1.  
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i.  

 
 

 

ii.  

♦♦♦ Mean mark data not available although “few” students were able to complete this part.
MARKER’S COMMENT: Better responses showed  ,  or    which eliminated  ½  from the algebra. Substituting  was a key element in completing this proof.
 
 
 
 
 
 

 

.

 

iii. 

MARKER’S COMMENT: Students must show sufficient working in proofs to convince markers they have derived the answer themselves.
 
 
 

Mechanics, EXT2* M1 2010 HSC 6b

A basketball player throws a ball with an initial velocity    m/s at an angle of    to the horizontal. At the time the ball is released its centre is at  , and the player is aiming for the point  as shown on the diagram. The line joining    and    makes an angle    with the horizontal, where  .
 

6b
 

Assume that at time    seconds after the ball is thrown its centre is at the point  ,  where

.      (DO NOT prove this.)

  1. If the centre of the ball passes through    show that
     
             (3 marks)

    (2) What happens to    as   ?    (1 mark)

  2. For a fixed value of  ,  let  .
     
    Show that     when    (2 marks)

  3. Using part (a)(ii)* or otherwise show that  ,   when  .    (1 mark)

     

    *Please note for the purposes of this question, part (a)(ii) showed that  when  ,  then   

  4. Explain why    is a minimum when       (2 marks)

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  1.  
  2. (1)  

     

    (2)  

  4.  
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i.   

♦ Mean mark 43%
IMPORTANT: It is critical to use the relationship  to achieve the required proof.

 

 
 
 

 

 

ii. (1) 

♦ Mean marks of 37% and 32% for parts (ii)(1) and (ii)(2) respectively.

ii. (2) 

 

iii.  .

 
 

 

♦♦ Mean mark part (iii) 21%
MARKER’S COMMENT: Many students failed to treat  as a constant when differentiating.

 

iv. 

 

♦♦♦ Mean mark 14%
MARKER’S COMMENT: Linking part (a)(ii) to this solution was a feature of the more efficient and successful approaches.

 

 

 

v.   

♦♦♦ Mean mark 9%
MARKER’S COMMENT: Only a small minority of students  explained correctly that    is a MIN when    is a MAX.