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Mechanics, EXT2 M1 2023 HSC 14c

A projectile of mass kg is launched vertically upwards from the origin with an initial speed m s. The acceleration due to gravity is ms.

The projectile experiences a resistive force of magnitude newtons, where is a positive constant and is the speed of the projectile at time seconds.

  1. The maximum height reached by the particle is metres.
  2. Show that  .  (3 marks)

  3. When the projectile lands on the ground, its speed is , where is less than the magnitude of the terminal velocity.
  4. Show that  .  (3 marks)

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i.   

 
 
 
 
 
 
   

 

 
   

 


  

ii.   

 
 
 
 
 
   

 


 

 
 
 
 
 
 
 
♦♦ Mean mark (ii) 31%.

Mechanics, EXT2 M1 2022 HSC 15a

A machine is lifted from the floor of a room using two ropes. The two ropes ensure that the horizontal components of the forces are balanced at all times. It is assumed that at all times the machine moves vertically upwards at a constant velocity.

The machine is located in a room with height metres.

One of the ropes is attached to the point on the machine and to the fixed point on the ceiling of the room. The point is a distance metres to the left of . Let the vertical distance from to the ceiling be metres and let be the angle this rope makes with the horizontal.

The other rope is attached to the point and to the fixed point on the floor of the room. The point is a distance metres to the right of . Let be the angle this rope makes with the horizontal.

Let the tension in the first rope be newtons, the tension in the second rope be newtons, the mass of the machine be kilograms and the acceleration due to gravity be .
 

  1. By considering horizontal and vertical components of the forces at , show that
  2.               (3 marks)

  3. Hence, or otherwise, show that the point cannot be lifted to a position metres above the floor.  (2 marks)

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i.   
       


 

 
   

 


Mean mark (i) 56%.

ii.   

 
   

 

 
 
 
 
 
 

 


♦♦♦ Mean mark (ii) 27%.

Mechanics, EXT2 M1 2021 HSC 15c

An object of mass 1 kg is projected vertically upwards with an initial velocity of m/s. It experiences air resistance of magnitude newtons where is the velocity of the object, in m/s, and is a positive constant. The height of the object above its starting point is metres. The time since projection is seconds and acceleration due to gravity is m/s².

  1. Show that the time for the object to reach its maximum height is    seconds.  (3 marks)

  2. Find an expression for the maximum height reached by the object, in terms of , , and .  (3 marks)

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i.   
 
   
   
   
 
   

 


  


 

ii.   

Mean mark part (ii) 54%.
 

 

 
 

Mechanics, EXT2 M1 2017 HSC 13c

A particle is projected upwards from ground level with initial velocity  , where is the acceleration due to gravity and is a positive constant. The particle moves through the air with speed    and experiences a resistive force.

The acceleration of the particle is given by  . Do NOT prove this.

The particle reaches a maximum height, , before returning to the ground.

Using  , or otherwise, show that    metres.  (4 marks)

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Mechanics, EXT2 M1 2009 HSC 7a

A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram. The jumper’s feet are tied to an elastic cord of length   m. The displacement of the jumper’s feet, measured downwards from the bridge, is   m.
 


 

The jumper’s fall can be examined in two stages. In the first stage of the fall, where  , the jumper falls a distance of   m subject to air resistance, and the cord does not provide resistance to the motion. In the second stage of the fall, where  , the cord stretches and provides additional resistance to the downward motion.

  1. The equation of motion for the jumper in the first stage of the fall is

     

          

     

    where    is the acceleration due to gravity,    is a positive constant, and    is the velocity of the jumper.
     
      (1)  Given that    and    initially, show that

             (3 marks)

      (2)  Given that    and  , find the length,  , of the cord such that the jumper’s velocity is    when  . Give your answer to two significant figures.  (1 mark)

  2. In the second stage of the fall, where  , the displacement    is given by
     
         
     
    where    is the time in seconds after the jumper’s feet pass  .

     

    Determine whether or not the jumper’s head stays out of the water.  (4 marks)

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i. (1)  
 
 

 

 
 

 

 
 

 

i. (2)   

 

 

ii.   
 
   
   

 

 
    

 

 

 
 

 

Mechanics, EXT2 M1 2013 HSC 15d

A ball of mass is projected vertically into the air from the ground with initial velocity  . After reaching the maximum height it falls back to the ground. While in the air, the ball experiences a resistive force , where is the velocity of the ball and is a constant.

The equation of motion when the ball falls can be written as

      (Do NOT prove this.)

  1. Show that the terminal velocity   of the ball when it falls is
  2.      (1 mark)

  3. Show that when the ball goes up, the maximum height    is
  4.      (3 marks)

  5. When the ball falls from height    it hits the ground with velocity  .
  6. Show that    (2 marks)

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i. 

 

ii. 

♦ Mean mark 43%.

MARKER’S COMMENT: More than half of students incorrectly wrote the equation to solve as
 
 
 

 

 

iii. 

♦♦ Mean mark 14%.