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Vectors, SPEC2 2023 VCAA 16 MC

A student throws a ball for his dog to retrieve. The position vector of the ball, relative to an origin at ground level seconds after release, is given by  . Displacement components are measured in metres, where is a unit vector to the east, is a unit vector to the north and is a unit vector vertically up.

The total distance, in metres, travelled by the ball before it hits the ground is closest to

  1. 1.5
  2. 11.5
  3. 13.0
  4. 24.5
  5. 26.0
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Vectors, SPEC2 2021 VCAA 4

A car that performs stunts moves along a track, as shown in the diagram below. The car accelerates from rest at point , is launched into the air by the ramp and lands on a second section of track at or beyond point .  This second section of track is inclined at to the horizontal.

Due to tailwind, the effect of air resistance is negligible. Point is taken as the origin of a cartesian coordinate system and all displacements are measured in metres. Point has the coordinates .

At point , the speed of the car is ms and it takes off at an angle of to the horizontal direction. After the car passes point , it follows a trajectory where the position of the car's rear wheels relative to point , is given by

  until the car lands on the second section of track that starts at point .
 

  1. Show that the path of the rear wheels of the car, while in the air, is given in cartesian form by
  2.    .   (1 mark)

  3. If  , find the minimum speed, in ms, that the car must reach at point for the rear wheels to land on the second section of track at or beyond point . Give your answer correct to two decimal places.   (2 marks)

  4. The ramp    is constructed so that the angle can be varied.
  5. For what values of and will the path of the rear wheels of the car join up smoothly with the beginning of the second section of track at point ? Give your answer for in degrees, correct to the nearest degree, and give your answer for in ms, correct to one decimal place.   (3 marks)

The car accelerates from rest along the horizontal section of track , where its acceleration,   ms, after it has travelled metres from point , is given by  , where is its speed at metres.

  1. Show that in terms of given by  .   (2 marks)

  2. After the car leaves point , it accelerates to reach a speed of 20 ms at point . However, if the stunt is called off, the car immediately brakes and reduces its speed at a rate of 9 ms.  It is only safe to call off the stunt if the car can come to rest at or before point .  Point is the furthest point along the section at which the stunt can be called off.
  3. How far is point from point ?  Give your answer in metres, correct to one decimal place.   (3 marks)

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a.   

 

 

b.   

♦ Mean mark part (b) 45%.

 

c.   

♦♦♦ Mean mark part (c) 12%.

 

d.   

♦ Mean mark part (d) 40%.

 

e.   

♦♦♦ Mean mark part (e) 8%.

 

 

 

 

Vectors, SPEC2-NHT 2019 VCAA 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. Show that the time spent in the air by the snowboarder is    seconds.   (3 marks)

  4. Find the total distance the snowboarder travels while airborne. Give your answer in metres, correct to two decimal places.   (2 marks)

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a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 


 
e.   

 

Vectors, SPEC2-NHT 2017 VCAA 4

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.   

 
 

  

 

 


 

Vectors, SPEC2-NHT 2018 VCAA 4

A basketball player aims to throw a basketball through a ring, the centre of which is at a horizontal distance of 4.5 m from the point of release of the ball and 3 m above floor level. The ball is released at a height of 1.75 m above floor level, at an angle of projection to the horizontal and at a speed of  . Air resistance is assumed to be negligible.
 


 

The position vector of the centre of the ball at any time, seconds, for  , relative to the point of release is given by 
 
,
 
where    is a unit vector in the horizontal direction of motion of the ball and    is a unit vector vertically up. Displacement components are measured in metres.

  1. For the player’s first shot at goal, and 
  2.   i. Find the time, in seconds, taken for the ball to reach its maximum height. Give your answer in the form  , where  and are positive integers.   (2 marks) 
  3.  ii. Find the maximum height, in metres, above floor level, reached by the centre of the ball.   (2 marks)

  4. iii. Find the distance of the centre of the ball from the centre of the ring one second after release. Give your answer in metres, correct to two decimal places.   (2 marks)

  5. For the player’s second shot at goal, .
    Find the possible angles of projection,  , for the centre of the ball to pass through the centre of the ring. Give your answers in degrees, correct to one decimal place.   (3 marks)

  6. For the player’s third shot at goal, the angle of projection is 
  7. Find the speed required for the centre of the ball to pass through the centre of the ring. Give your answer in metres per second, correct to one decimal place.   (2 marks)

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a.i.   
   
     

 

 

a.ii.  
   

 

 

a.iii. 

 
 
   
   

 

b.  


 

 

c.