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Calculus, 2ADV C3 2021 HSC 26

A particle is shot vertically upwards from a point 100 metres above ground level.

The position of the particle, metres above the ground after seconds, is given by

.

  1. Find the maximum height above ground level reached by the particle.  (2 marks)

  2. Find the velocity of the particle, in metres per second, immediately before it hits the ground, leaving your answer in the form  ,  where    and    are integers.  (3 marks)

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a.   

 

 

 

b.   

♦ Mean mark 38%.

 

 
 

 

 

 

Calculus, 2ADV C3 2004 HSC 5b

A particle moves along a straight line so that its displacement, metres, from a fixed point is given by  , where    is measured in seconds.

  1. What is the initial displacement of the particle?  (1 mark)

  2. Sketch the graph of    as a function of    for  .  (2 marks)

  3. Hence, or otherwise, find when AND where the particle first comes to rest after  .  (2 marks)

  4. Find a time when the particle reaches its greatest magnitude of velocity. What is this velocity?  (2 marks)

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i.   

 
 

 

 

 ii. 

 

 Calculus in the Physical World, 2UA 2004 HSC 5b

 

iii.  
 

 

 

 
 
 

 

 

iv. 

   

 

 

 
 

 

Calculus, 2ADV C4 2009 HSC 7a

The acceleration of a particle is given by

,

where    is the displacement in metres and    is the time in seconds.

Initially its velocity is  and its displacement is 5 m.

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

  3. Find the time when the particle comes to rest.    (3 marks)

  4. Find the displacement  when the particle comes to rest.    (1 mark)

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i.   

 

 

 
 

 

 

 

ii.  

 
 

 


 


  

iii. 

ALGEBRA TIP: Helpful identity  . Easily provable as follows:



.

Calculus, 2ADV C3 2011 HSC 7b

The velocity of a particle moving along the -axis is given by

,

where is the time in seconds and is the displacement in metres.

  1. Show that the particle is initially at rest.     (1 mark)

  2. Show that the acceleration of the particle is always positive.     (1 mark)

  3. Explain why the particle is moving in the positive direction for all  .     (2 marks)

  4. As  , the velocity of the particle approaches a constant.

     

    Find the value of this constant.     (1 mark) 

  5. Sketch the graph of the particle's velocity as a function of time.     (2 marks)

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i.   

 
 

 

 

MARKER’S COMMENT: Students whose working showed as , tended to score highly in this question.

ii.   

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.
 

.
 

iii. 

♦♦♦ Mean mark 22%
COMMENT: Students found part (iii) the most challenging part of this question by far.


 

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iv.   ,  


 

 

IMPORTANT: Use previous parts to inform this diagram. i.e. clearly show velocity was zero at    and the asymptote at  . 
v.   

Calculus in the Physical World, 2UA 2011 HSC 7b Answer