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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 2019 HSC 13d

The point    is on a sloping plane that forms an angle of 45° to the horizontal. A particle is projected from the point  . The particle hits a point    on the sloping plane as shown in the diagram.
 


 

The equation of the line    is  . The equations of motion of the particle are

where    is the time in seconds after projection. Do NOT prove these equations.

  1. Find the distance    between the point of projection and the point where the particle hits the sloping plane.  (2 marks)

  2. What is the size of the acute angle that the path of the particle makes with the sloping plane as the particle hits the point  ?  (3 marks)

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i.   
 

 

 

 
 

 

ii.   
 

 

 
 
 

 

♦ Mean mark part (ii) 39%.
 

 

 
 

 

Mechanics, EXT2* M1 2018 HSC 13c

An object is projected from the origin with an initial velocity of  at an angle    to the horizontal. The equations of motion of the object are

  (Do NOT prove this.)

 

  1. Show that when the object is projected at an angle  , the horizontal range is

     

           (2 marks)

  2. Show that when the object is projected at an angle  , the horizontal range is also 

     

         .  (1 mark)

  3. The object is projected with initial velocity to reach a horizontal distance , which is less than the maximum possible horizontal range. There are two angles at which the object can be projected in order to travel that horizontal distance before landing.

     

    Let these angles be   and  , where 

     

    Let    be the maximum height reached by the object when projected at the angle to the horizontal.

     

    Let    be the maximum height reached by the object when projected at the angle to the horizontal.
     
         
     
    Show that the average of the two heights, , depends only on and .  (3 marks)

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i.   

 

 
 
 

 

ii. 

 
 

 

iii. 


  

 
 
 

 

♦ Mean mark 44%.
 

 
 
 
 

 

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 2016 HSC 13b

The trajectory of a projectile fired with speed    at an angle    to the horizontal is represented by the parametric equations

  and   ,

where is the time in seconds.

  1. Prove that the greatest height reached by the projectile is  .  (2 marks)

A ball is thrown from a point above the horizontal ground. It is thrown with speed at an angle of to the horizontal. At its highest point the ball hits a wall, as shown in the diagram.
 

     ext1-2016-hsc-q13
 

  1. Show that the ball hits the wall at a height of above the ground.  (2 marks)

The ball then rebounds horizontally from the wall with speed . You may assume that the acceleration due to gravity is .

  1. How long does it take the ball to reach the ground after it rebounds from the wall?  (2 marks)

  2. How far from the wall is the ball when it hits the ground?  (1 mark)

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i.   
 

 

 

 

ii.  

 

♦♦ Mean mark part (iii) 35%.
iii.   ext1-hsc-2016-13bi

 

MARKER’S COMMENT: Many students struggled to solve: .

 

 

iv.  

♦ Mean mark 43%.

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 2015 HSC 14a

A projectile is fired from the origin with initial velocity m s at an angle to the horizontal. The equations of motion are given by

.    (Do NOT prove this)
 

Calculus in the Physical World, EXT1 2015 HSC 14a
 

  1. Show that the horizontal range of the projectile is
     
         .  (2 marks)

A particular projectile is fired so that  .

  1. Find the angle that this projectile makes with the horizontal when
     
         .  (2 marks)


  2. State whether this projectile is travelling upwards or downwards when
     
          . Justify your answer.  (1 mark)

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i.   

 

 

 

 

♦♦ Mean mark part (ii) 28%.
ii.   
 
 
 

 

 
 
 

Calculus in the Physical World, EXT1 2015 HSC 14a Answer

α
 
 
 
α

 

 

iii. 

♦♦ Mean mark part (iii) 31%.

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 2011 HSC 6b

The diagram shows the trajectory of a ball thrown horizontally, at speed m/s, from the top of a tower metres above the ground level.
 

2011 6b
 

The ball strikes the ground at an angle of  45°, metres from the base of the tower, as shown in the diagram. The equations describing the trajectory of the ball are

  and   ,    (DO NOT prove this)

where is the acceleration due to gravity, and is time in seconds.

  1. Prove that the ball strikes the ground at time  
     
           seconds.    (1 mark)

  2. Hence, or otherwise, show that  .    (2 marks)

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  1.  
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i.  

 

ii.  

,

 

 

♦♦ Mean mark 33%
STRATEGY: The key to unlocking this proof is understanding that , where   is the angle of trajectory at impact of a projectile, equals  .