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Calculus, 2ADV C4 2023 HSC 26

A camera films the motion of a swing in a park.

Let be the horizontal distance, in metres, from the camera to the seat of the swing at seconds.

The seat is released from rest at a horizontal distance of 11.2 m from the camera.
 

  1. The rate of change of can be modelled by the equation

.

  1. Find an expression for .  (2 marks)

  2. How many times does the swing reach the closest point to the camera during the first 10 seconds?  (2 marks)

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a.   

 
   
   
Mean mark (a) 51%.

  

 
 
   

 

 

b.   
♦♦ Mean mark (b) 31%.

 
   

 

Calculus, 2ADV C4 EQ-Bank 5

The velocity of a particle moving along the -axis at metres per second at seconds, is shown in the graph below.
 


 

Initially, the displacement is equal to 12 metres.

  1. Write an equation that describes the displacement, , at time seconds.  (2 marks)

  2. Draw a graph that shows the displacement of the particle,   metres from the origin, at a time seconds between    and  . Label the coordinates of the endpoints of your graph.  (2 marks)

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  2.  

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ii. 


 

Calculus, 2ADV C4 SM-Bank 1 MC

A lift accelerates from rest at a constant rate until it reaches a speed of 3 ms−1. It continues at this speed for 10 seconds and then decelerates at a constant rate before coming to rest. The total travel time for the lift is 30 seconds.

The total distance, in metres, travelled by the lift is

  1.  45
  2.  60
  3.  75
  4.  90
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Calculus, 2ADV C4 2007* HSC 10a

An object is moving on the -axis. The graph shows the velocity, , of the object, as a function of time, . The coordinates of the points shown on the graph are  . The velocity is constant for  .
 


 

  1. The object is initially at the origin. During which time(s) is the displacement of the object decreasing?  (1 mark)

  2. If the object travels 7 units in the first 4 seconds, estimate the time at which the object returns to the origin. Justify your answer.  (2 marks)

  3. Sketch the displacement, , as a function of time.  (2 marks)

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  3.    
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Calculus, 2ADV C4 2016 HSC 16a

A particle moves in a straight line. Its velocity at time seconds is given by

  1. Find the initial velocity.  (1 mark)

  2. Find the acceleration of the particle when the particle is stationary.  (2 marks)

  3. By considering the behaviour of for large , sketch a graph of against for  , showing any intercepts.  (2 marks)

  4. Find the exact distance travelled by the particle in the first 7 seconds.  (3 marks)

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  3.  
    hsc-2016-16ai
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i.  

Mean mark 93%.
COMMENT: Great example of low hanging fruit late in the exam for students who progress efficiently.

 

ii.  
 

 

   

   

 

iii. 

♦ Mean mark 47%.

 hsc-2016-16ai

 

♦♦ Mean mark 26%.

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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ii.  

 
 

 


 


  

iii. 

ALGEBRA TIP: Helpful identity  . Easily provable as follows:



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Calculus, 2ADV C4 2012 HSC 15b

The velocity of a particle is given by

,

where    is the displacement in metres and    is the time in seconds. Initially the particle is 3 m to the right of the origin.

  1. Find the initial velocity of the particle.    (1 mark)

  2. Find the maximum velocity of the particle.    (1 mark)

  3. Find the displacement, ,  of the particle in terms of  .    (2 marks)

  4. Find the position of the particle when it is at rest for the first time.    (2 marks)

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i.    :

 
 

 

 

ii.  

♦♦ Mean mark 29%
MARKER’S COMMENT: Solution 2 is more efficient here. Using the -1 and +1 limits of trig functions can be very a effective way to calculate max/min values.


 

,  ,  , …

 

 

 

 

iii.   
 
 

 

 

 

iv.  

♦ Mean mark 50%
MARKER’S COMMENT: Many students found    but failed to gain full marks by omitting to find  . Remember that for calculus, angles are measured in radians, NOT degrees!