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Mechanics, EXT2 M1 2023 SPEC1 8

A body moves in a straight line so that when its displacement from a fixed origin is metres, its acceleration, , is . The body accelerates from rest and its velocity, , is equal to as it passes through the origin. The body then comes to rest again.

Find in terms of for this interval.  (4 marks)

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Mechanics, EXT2* M1 2019 HSC 13c

A particle moves in a straight line. At time    seconds the particle has a displacement of m, a velocity of    and acceleration  .

Initially the particle has displacement  0 m  and velocity  . The acceleration is given by  . The velocity of the particle is always positive.

  1. Show that  .  (2 marks)

  2. Find an expression for as a function of  .   (2 marks)

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i.   
 
 
   

 

 

 

 

ii.   
 
 
   
   

 

♦ Mean mark part (ii) 46%.

Mechanics, EXT2 M1 2016 HSC 15b

A particle is initially at rest at the point which is metres to the right of

The particle then moves in a straight line towards

For the acceleration of the particle is given by    where is the distance from and is a positive constant.

  1. Prove that    (2 marks)

  2. Using the substitution   show that the time taken to reach a distance metres to the right of is given by
     
           (3 marks)

It can be shown that    (Do NOT prove this.)

  1. What is the limiting time taken for the particle to reach   (1 mark)

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i.  

 

 

 

 

ii.   
 
 

 

 

 

 

 

 

iii.  

 

♦♦ Mean mark 29%.
 
 

Mechanics, EXT2 M1 2006 HSC 6b

In an alien universe, the gravitational attraction between two bodies is proportional to  , where    is the distance between their centres.

A particle is projected upward from the surface of a planet with velocity    at time  .  Its distance    from the centre of the planet satisfies the equation

  1. Show that  , where    is the magnitude of the acceleration due to gravity at the surface of the planet and    is the radius of the planet.  (1 mark)

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

  3. It can be shown that   (Do NOT prove this.)

     

    Show that if    the particle will not return to the planet.  (2 marks)

  4. If    the particle reaches a point whose distance from the centre of the planet is  , and then falls back.

  5.  

      (1)   Use the formula in part (ii) to find    in terms of    and    (1 mark)

  6.  

      (2)   Use the formula in part (iii) to find the time taken for the particle to return to the surface of the planet in terms of    and    (1 mark)

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  4.  

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ii.   
 
   
 

 

 

 

iii. 

 
 
 
 

 

 

 

 

 

iv. (1)  

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iv. (2)  

 

Mechanics, EXT2* M1 2007 HSC 3c

A particle is moving in a straight line with its acceleration as a function of    given by  . It is initially at the origin and is travelling with a velocity of 1 metre per second.

  1. Show that  .  (2 marks)

  2. Hence show that  .  (2 marks)

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i.   

 

 

 

MARKER’S COMMENT: Most candidates “neglected” to consider the two cases that  .

 

 

ii.   

 

 

Mechanics, EXT2* M1 2004 HSC 5a

A particle is moving along the -axis, starting from a position    metres to the right of the origin (that is,    when  ) with an initial velocity of    and an acceleration given by

.

  1. Show that  .  (2 marks)

  2. Hence find an expression for    in terms of  .  (3 marks)

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i.   

 

 

 
 

  

ii.   
 
 
   

 

 

Mechanics, EXT2* M1 2006 HSC 4c

A particle is moving so that  

Initially    and the velocity, , is  

  1. Show that    (2 marks)

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

  3. It can be shown that for some constant  ,
     
                (Do NOT prove this.)
     
    Using this equation and the initial conditions, find    as a function of    (2 marks)

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i.   

 
 

 

 

 
 

 

ii.   


 

 

 

 

iii. 

 

 

 
 
 

Mechanics, EXT2* M1 2015 HSC 14b

A particle is moving horizontally. Initially the particle is at the origin moving with velocity .

The acceleration of the particle is given by  , where is its displacement at time  .

  1. Show that the velocity of the particle is given by  .  (3 marks)

  2. Find an expression for as a function of .  (2 marks)

  3. Find the limiting position of the particle.  (1 mark)

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i.   

 
 

 

 

 

 

 

ii.  
 
 
   

 

♦ Mean mark 49%.

 

♦ Mean mark 42%.
iii.