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Vectors, EXT1 V1 2022 HSC 14c

A video game designer wants to include an obstacle in the game they are developing. The player will reach one side of a pit and must shoot a projectile to hit a target on the other side of the pit in order to be able to cross. However, the instant the player shoots, the target begins to move away from the player at a constant speed that is half the initial speed of the projectile shot by the player, as shown in the diagram below.

The initial distance between the player and the target is , the initial speed of the projectile is and it is launched at an angle of to the horizontal. The acceleration due to gravity is . The launch angle is the ONLY parameter that the player can change.
 

  
 

Taking the position of the player when the projectile is launched as the origin, the positions of the projectile and target at time after the projectile is launched are as follows.

     

 
Show that, for the player to have a chance of hitting the target, must be less than 37% of the maximum possible range of the projectile (to 2 significant figures).  (4 marks)

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♦♦ Mean mark 33%.

Vectors, EXT1 V1 SM-Bank 30

A canon ball is fired from a castle wall across a horizontal plane at ms−1.

Its position vector  seconds after it is fired from its origin is given by  .

  1. If the projectile hits the ground at a distance 8 times the height at which it was fired, show that it initial velocity is given by
     
            (2 marks)

  2. Show that the total distance the canon ball travels can be expressed as
     
                (2 marks)

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i.   

 

 

 

ii.   

 
 

 

 

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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  1.  

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a.   


 

 
   
 

 

b. 

 

 

 

 

c.   


 

d. 

 
 


e.