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Vectors, EXT1 V1 2024 HSC 14d

A particle is projected from the origin, with initial speed at an angle of to the horizontal. The position vector of the particle, , where is the time after projection and is the acceleration due to gravity, is given by

.   (Do NOT prove this.)

Let be the distance of the particle from the origin at time , so  .

Show that for    the distance, , is increasing for all  .   (4 marks)

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♦♦♦ Mean mark 25%.

 
   


 

 
 
 
 
 

Vectors, EXT1 V1 2019 SPEC2-N 4

A snowboarder at the Winter Olympics leaves a ski jump at an angle of degrees to the horizontal, rises up in the air, performs various tricks and then lands at a distance down a straight slope that makes an angle of 45° to the horizontal, as shown below.

Let the origin of a cartesian coordinate system be at the point where the snowboarder leaves the jump, with a unit vector in the positive direction being represented by    and a unit vector in the positive direction being represented by  . Distances are measured in metres and time is measured in seconds.

The position vector of the snowboarder    seconds after leaving the jump is given by


 


 

  1. Find the angle  .    (2 marks)

  2. Find the speed, in metres per second, of the snowboarder when she leaves the jump at .    (1 mark)

  3. Find the maximum height above reached by the snow boarder. Give your answer in metres, correct to one decimal place.    (2 marks)

  4. Show that the time spent in the air by the snowboarder is    seconds.    (3 marks)

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a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 

Vectors, EXT1 V1 SM-Bank 6

A cricketer hits a ball at time    seconds from an origin at ground level across a level playing field.

The position vector  , from , of the ball after seconds is given by
 
  ,
 
where,    is a unit vector in the forward direction,   is a unit vector vertically up and displacement components are measured in metres.

  1. Find the initial velocity of the ball and the initial angle, in degrees, of its trajectory to the horizontal.  (2 marks)

  2. Find the maximum height reached by the ball, giving your answer in metres, correct to two decimal places.  (2 marks)

  3. Find the time of flight of the ball. Give your answer in seconds, correct to three decimal places.  (1 mark)

  4. Find the range of the ball in metres, correct to one decimal place.  (1 mark)

  5. A fielder, more than 40 m from , catches the ball at a height of 2 m above the ground.

     

    How far horizontally from is the fielder when the ball is caught? Give your answer in metres, correct to one decimal place.  (2 marks)

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a.   


 

 
   
 

 

b. 

 

 

 

 

c.   


 

d. 

 
 


e.