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Calculus, EXT1 C1 EQ-Bank 2

A particle is moving along the -axis where its displacement, in metres from the origin, after seconds, is given by:

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  1. Determine the times when the particle is at rest.   (2 marks)

  2. Find and describe the velocity of the particle at the time where there is no accelerating force acting on it.  (2 marks)

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Calculus, EXT1* C1 2019 HSC 10 MC

A particle is moving along a straight line with displacement at time .

The particle is stationary when    and when  .

Which of the following MUST be true in this case?

A.  The particle changes direction at some time between    and  .
B.   The displacement function of the particle has a stationary point at some time between    and  .
C.   The acceleration of the particle is 0 at some time between    and  .
D.   The acceleration function of the particle has a stationary point at some time between    and  .
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♦ Mean mark 37%.

Calculus, EXT1* C1 2017 HSC 15c

Two particles move along the -axis.

When  , particle is at the origin and moving with velocity 3.

For  , particle has acceleration given by  .

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

When  , particle is also at the origin.

For  , particle has velocity given by  .

  1. When do the two particles have the same velocity?  (2 marks)

  2. Show that the two particles do not meet for  .  (3 marks)

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♦ Mean mark (iii) 39%.

 

 

 

 

 

 

Calculus, EXT1* C1 2004 HSC 9b

A particle moves along the -axis. Initially it is at rest at the origin. The graph shows the acceleration, , of the particle as a function of time    for  .
 

2004 9b
 

  1. Write down the time at which the velocity of the particle is a maximum.  (1 marks)

  2. At what time during the interval    is the particle furthest from the origin? Give brief reasons for your answer.  (2 marks)

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Calculus, EXT1* C1 2007 HSC 5b

A particle is moving on the -axis and is initially at the origin. Its velocity, metres per second, at time seconds is given by

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

  2. Find an expression for the acceleration of the particle.  (2 marks)

  3. Find the time when the acceleration of the particle is zero.  (1 mark)

  4. Find the position of the particle when .  (3 marks)

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Calculus, EXT1* C1 2015 HSC 14a

In a theme park ride, a chair is released from a height of    metres and falls vertically. Magnetic brakes are applied when the velocity of the chair reaches    metres per second.
 

2015 2ua 14a
 

The height of the chair at time seconds is metres. The acceleration of the chair is given by   . At the release point,  .

  1. Using calculus, show that  .  (2 marks)

  2. How far has the chair fallen when the magnetic brakes are applied?  (2 marks)

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Calculus, EXT1* C1 2005 HSC 9a

A particle is initially at rest at the origin. Its acceleration as a function of time, , is given by

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

  2. Sketch the graph of the velocity for  π  AND determine the time at which the particle first comes to rest after  .  (3 marks)

  3. Find the distance travelled by the particle between    and the time at which the particle first comes to rest after  .  (2 marks)

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Calculus in the Physical World, 2UA 2005 HSC 9a Answer

 

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Calculus, EXT1* C1 2005 HSC 7b

Calculus in the Physical World, 2UA 2005 HSC 7b
 

The graph shows the velocity, , of a particle as a function of time. Initially the particle is at the origin. 

  1. At what time is the displacement, , from the origin a maximum?  (1 mark)

  2. At what time does the particle return to the origin? Justify your answer.  (2 marks)

  3. Draw a sketch of the acceleration,  , as a function of time for  .  (2 marks)

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iii.   Calculus in the Physical World, 2UA 2005 HSC 7b Answer

Calculus, EXT1* C1 2013 HSC 10 MC

A particle is moving along the  -axis. The displacement of the particle at time    seconds is    metres.

At a certain time,    and  .

Which statement describes the motion of the particle at that time?

  1. The particle is moving to the right with increasing speed.
  2. The particle is moving to the left with increasing speed.
  3. The particle is moving to the right with decreasing speed.
  4. The particle is moving to the left with decreasing speed.
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♦♦ Mean mark 26%.

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Calculus, EXT1* C1 2010 HSC 7a

The acceleration of a particle is given by

,

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

Initially the particle is at the origin with a velocity of  .

  1. Show that the velocity of the particle is given by

     

      .    (2 marks)

  2. Find the time when the particle first comes to rest.    (2 marks)

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

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Calculus, EXT1* C1 2013 HSC 14a

The velocity of a particle moving along the -axis is given by  , where is the displacement from the origin in metres and is the time in seconds. Initially the particle is 5 metres to the right of the origin.

  1. Show that the acceleration of the particle is constant.  (1 mark)

  2. Find the time when the particle is at rest.  (1 mark)

  3. Show that the position of the particle after 7 seconds is 26 metres to the right of the origin.  (2 marks)

  4. Find the distance travelled by the particle during the first 7 seconds.   (2 marks)

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♦ Mean mark 37%.
IMPORTANT: Students can also draw a number line that shows where a particle starts from, changes direction and finishes in order to reduce errors.

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