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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 2005 6b

An experimental rocket is at a height of  5000 m, ascending with a velocity of    at an angle of  45°  to the horizontal, when its engine stops.
 

 
After this time, the equations of motion of the rocket are:

where is measured in seconds after the engine stops. (Do NOT show this.)

  1. What is the maximum height the rocket will reach, and when will it reach this height?  (2 marks)

  2. The pilot can only operate the ejection seat when the rocket is descending at an angle between 45° and 60° to the horizontal. What are the earliest and latest times that the pilot can operate the ejection seat?  (3 marks)

  3. For the parachute to open safely, the pilot must eject when the speed of the rocket is no more than  . What is the latest time at which the pilot can eject safely?  (2 marks)

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  1.  

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i.

    

 

 

 
 

 

 

ii. 

 

 

 

 
 
 

 

 

iii.

 
 

 

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 2013 HSC 13c

Points and are located metres apart on a horizontal plane. A projectile is fired from towards with initial velocity m/s at angle to the horizontal.

At the same time, another projectile is fired from towards with initial velocity m/s at angle to the horizontal, as shown on the diagram.

The projectiles collide when they both reach their maximum height.
 

2013 13c
 

The equations of motion of a projectile fired from the origin with initial velocity  m/s at angle    to the horizontal are

  and   .        (DO NOT prove this.)

  1. How long does the projectile fired from    take to reach its maximum height?  (2 marks)

  2. Show that  .    (1 mark)

  3. Show that  .    (2 marks)

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i.   

 

 

 

 

ii.  

IMPORTANT: Part (i) in this example leads students to a very quick solution. Always look to previous parts for clues to direct your strategy.

 

 

iii 

 
 

 

♦ Mean mark 39%
IMPORTANT: Students should direct their calculations by reverse engineering the required result. in the proof means that   will appear in the working calculations.