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Mechanics, EXT2* M1 2019 HSC 13d

The point    is on a sloping plane that forms an angle of 45° to the horizontal. A particle is projected from the point  . The particle hits a point    on the sloping plane as shown in the diagram.
 


 

The equation of the line    is  . The equations of motion of the particle are

where    is the time in seconds after projection. Do NOT prove these equations.

  1. Find the distance    between the point of projection and the point where the particle hits the sloping plane.  (2 marks)

  2. What is the size of the acute angle that the path of the particle makes with the sloping plane as the particle hits the point  ?  (3 marks)

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i.   
 

 

 

 
 

 

ii.   
 

 

 
 
 

 

♦ Mean mark part (ii) 39%.
 

 

 
 

 

Mechanics, EXT2* M1 2005 6b

An experimental rocket is at a height of  5000 m, ascending with a velocity of    at an angle of  45°  to the horizontal, when its engine stops.
 

 
After this time, the equations of motion of the rocket are:

where is measured in seconds after the engine stops. (Do NOT show this.)

  1. What is the maximum height the rocket will reach, and when will it reach this height?  (2 marks)

  2. The pilot can only operate the ejection seat when the rocket is descending at an angle between 45° and 60° to the horizontal. What are the earliest and latest times that the pilot can operate the ejection seat?  (3 marks)

  3. For the parachute to open safely, the pilot must eject when the speed of the rocket is no more than  . What is the latest time at which the pilot can eject safely?  (2 marks)

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  1.  

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ii. 

 

 

 

 
 
 

 

 

iii.

 
 

 

Mechanics, EXT2* M1 2015 HSC 14a

A projectile is fired from the origin with initial velocity m s at an angle to the horizontal. The equations of motion are given by

.    (Do NOT prove this)
 

Calculus in the Physical World, EXT1 2015 HSC 14a
 

  1. Show that the horizontal range of the projectile is
     
         .  (2 marks)

A particular projectile is fired so that  .

  1. Find the angle that this projectile makes with the horizontal when
     
         .  (2 marks)


  2. State whether this projectile is travelling upwards or downwards when
     
          . Justify your answer.  (1 mark)

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i.   

 

 

 

 

♦♦ Mean mark part (ii) 28%.
ii.   
 
 
 

 

 
 
 

Calculus in the Physical World, EXT1 2015 HSC 14a Answer

α
 
 
 
α

 

 

iii. 

♦♦ Mean mark part (iii) 31%.

Mechanics, EXT2* M1 2011 HSC 6b

The diagram shows the trajectory of a ball thrown horizontally, at speed m/s, from the top of a tower metres above the ground level.
 

2011 6b
 

The ball strikes the ground at an angle of  45°, metres from the base of the tower, as shown in the diagram. The equations describing the trajectory of the ball are

  and   ,    (DO NOT prove this)

where is the acceleration due to gravity, and is time in seconds.

  1. Prove that the ball strikes the ground at time  
     
           seconds.    (1 mark)

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

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  1.  
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i.  

 

ii.  

,

 

 

♦♦ Mean mark 33%
STRATEGY: The key to unlocking this proof is understanding that , where   is the angle of trajectory at impact of a projectile, equals  .