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Mechanics, EXT2* M1 2019 HSC 12b

A particle is moving along the -axis in simple harmonic motion. The position of the particle is given by

for   

  1. Write    in the form  , where    and  .  (2 marks)

  2. Find the two values for    where the particle comes to rest.   (1 mark)

  3. When is the first time that the speed of the particle is equal to half of its maximum speed?  (2 marks)

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i.   

  

 

♦ Mean mark part (ii) 46%.

 

ii.   
 

 

iii.   

♦ Mean mark part (iii) 49%.

 

 

 

Mechanics, EXT2* M1 2007 HSC 6a

A particle moves in a straight line. Its displacement, metres, after seconds is given by

.

  1. Prove that the particle is moving in simple harmonic motion about    by showing that   .  (2 marks)

  2. What is the period of the motion?  (1 mark)

  3. Express the velocity of the particle in the form  α, where  α  is in radians.  (2 marks)

  4. Hence, or otherwise, find all times within the first    seconds when the particle is moving at metres per second in either direction.  (2 marks)

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i.   

 
 
 

 

ii. 

 

 

iii. 

α

αα
αα
α
α
 

 

α
α

 

(iv) 

 

 

Mechanics, EXT2* M1 2005 HSC 5c

A particle moves in a straight line and its position at time    is given by

  1. Express    in the form    where    is in radians.  (2 marks)

  2. The particle is undergoing simple harmonic motion. Find the amplitude and the centre of the motion.  (2 marks)

  3. When does the particle first reach its maximum speed after time  ?  (1 mark)

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i.   

 

 

 

 

ii. 
 

 

 

iii. 

 

 

Mechanics, EXT2* M1 2012 HSC 13c

A particle is moving in a straight line according to the equation

, 

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

  1. Prove that the particle is moving in simple harmonic motion by showing that  satisfies an equation of the form  .  (2 marks)

  2. When is the displacement of the particle zero for the first time?    (3 marks)

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i. 

 
 
 

 

ii. 

♦ Mean mark 42%.
IMPORTANT: The critical insight required to solve is to realise that the cosine of the difference between 2 angles, i.e. , applies.

 

 Calculus in the Physical World, EXT1 2012 HSC 13c Answer