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Mechanics, EXT2 M1 2024 HSC 16c

Two particles, and , each have mass 1 kg and are in a medium that exerts a resistance to motion equal to , where    and is the velocity of any particle. Both particles maintain vertical trajectories.

The acceleration due to gravity is ms, where  .

The two particles are simultaneously projected towards each other with the same speed, ms, where  .

The particle is initially metres directly above particle , where  .

Find the time taken for the particles to meet.   (4 marks)

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♦♦♦ Mean mark 28%.

 
 
 
 

 
   
   

 


 


 

 
 

 


 

 
 
 
 

Mechanics, EXT2 M1 2024 HSC 15c

A bar magnet is held vertically. An object that is repelled by the magnet is to be dropped from directly above the magnet and will maintain a vertical trajectory. Let be the distance of the object above the magnet.
 

The object is subject to acceleration due to gravity, , and an acceleration due to the magnet , so that the total acceleration of the object is given by

 

The object is released from rest at  .

  1. Show that  .   (2 marks)

  2. Find where the object next comes to rest, giving your answer correct to 1 decimal place.  (2 marks)

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♦♦♦ Mean mark (ii) 24%.

 
   
   
   

 
 

Mechanics, EXT2 M1 2024 HSC 13c

A particle of unit mass moves horizontally in a straight line. It experiences a resistive force proportional to , where m s is the speed of the particle, so that the acceleration is given by  .

Initially the particle is at the origin and has a velocity of 40 m s to the right. After the particle has moved 15 m to the right, its velocity is 10 m s (to the right).

  1. Show that  .   (3 marks)

  2. Show that  .   (1 mark)

  3. At what time will the particle's velocity be 30 m s to the right?   (3 marks)

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Mechanics, EXT2 M1 EQ-Bank 4

A torpedo with a mass of 80 kilograms has a propeller system that delivers a force of on the torpedo, at maximum power. The water exerts a resistance on the torpedo proportional to the square of the torpedo's velocity .

  1. Explain why 
     
    where is a positive constant.  (1 mark)

  2. If the torpedo increases its velocity from    to  , show that the distance it travels in this time, , is given by
     
           (3 marks)

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Mechanics, EXT2 M1 2015 HSC 15a

A particle   of unit mass travels horizontally through a viscous medium. When  , the particle is at point    with initial speed  . The resistance on particle    due to the medium is  , where    is the velocity of the particle at time    and    is a positive constant.

When  , a second particle    of equal mass is projected vertically upwards from    with the same initial speed    through the same medium. It experiences both a gravitational force and a resistance due to the medium. The resistance on particle    is  , where    is the velocity of the particle    at time  . The acceleration due to gravity is  .

  1. Show that the velocity    of particle    is given by  
     
           (2 marks)

  2. By considering the velocity    of particle  , show that
     
           (3 marks)

  3. Show that the velocity    of particle    when particle    is at rest is given by
     
           (1 mark)

  4. Hence, if    is very large, explain why  
     
           (1 mark)

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Mechanics, EXT2 M1 2012 HSC 13a

An object on the surface of a liquid is released at time    and immediately sinks. Let    be its displacement in metres in a downward direction from the surface at time    seconds.

The equation of motion is given by

,

where    is the velocity of the object.  

  1. Show that  .   (4 marks)

  2. Use   to show that  
     
            (2 marks)

     

  3. How far does the object sink in the first 4 seconds?   (2 marks)

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