DP IB Maths: AI HL

Topic Questions

3.9 Modelling with Vectors

1a
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4 marks

Two spaceships straight A space and space straight B, in a 3D virtual reality game, are moving such that their positions relative to a fixed point straight O at time t seconds, 0 less or equal than t less than 30, are defined by the position vectors r subscript A equals open parentheses table row 2 row cell negative 3.5 end cell row 1 end table close parentheses plus t open parentheses table row cell 1.2 end cell row cell 0.5 end cell row 2 end table close parentheses and r subscript B equals open parentheses table row cell negative 2 end cell row 4 row cell 9.5 end cell end table close parentheses plus t open parentheses table row 2 row cell negative 1 end cell row cell 0.3 end cell end table close parentheses respectively.

Show that the two spaceships are on course to collide at point straight P and write down the coordinates of straight P.

1b
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1 mark

Spaceship straight B reduces its velocity such that its position vector is now given by

r subscript B equals open parentheses table row cell negative 2 end cell row 4 row cell 9.5 end cell end table close parentheses plus t open parentheses table row cell 1.6 end cell row cell negative 0.8 end cell row cell 0.24 end cell end table close parentheses 

Show that spaceship B is still travelling in its original direction.

1c
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5 marks

Show that the distance between the two spaceships can be written as

square root of 4.9476 t squared minus 56.62 t plus 144.5 end root
1d
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2 marks

Hence find the distance between the two spaceships when spaceship straight A is at straight P.

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2a
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2 marks

A car is moving with constant velocity along the line with equation r subscript c equals open parentheses table row 2 row 3 end table close parentheses plus t open parentheses table row 5 row 12 end table close parentheses. A bird is perched at the point open parentheses 25 comma space 32 comma 8 close parentheses and at t equals 0 , starts to fly at a constant velocity in the direction of the vector open parentheses 2 straight i plus 31 straight j minus 4 k close parentheses.

All distances are measured in metres and time in seconds. The base vectors straight i and straight j represent due east and due west respectively and the base vector bold italic k points upwards.

Verify that the bird does not collide with the car.
2b
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3 marks

Show that at some point in time the bird will be directly above the car and state the time at which this occurs.

2c
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2 marks

Hence find the distance between the bird and the car at that time.

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3a
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4 marks

A hawk is flying with constant velocity, v, measured in kilometres per minute, where

 v equals open parentheses table row cell negative 1 end cell row a row cell 0.2 end cell end table close parentheses

A fixed public telescope is located at a point A open parentheses 1 comma negative 2 close parentheses relative to an origin O.  At time t equals 0 minutes the hawk is at the point open parentheses 4 comma negative 1 comma 1 close parentheses and at time t equals T minutes the hawk is vertically above the telescope.

The xdirection is due east, the y direction is due north and the z direction is vertically upwards. All distances are measured in kilometres. 

Find the value of a and the height of the hawk when it is directly above the telescope.

3b
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6 marks

Given that the hawk continues flying in the same direction, find 

i)
the speed at which the hawk is flying,
ii)
the time at which the hawk is exactly 4 km from the telescope.

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4a
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3 marks

An eagle is tracking a mouse on the ground that is moving with constant velocity such that its position vector relative to an origin O, at time t  seconds can be modelled by

 open parentheses table row x row y end table close parentheses equals open parentheses table row 30 row cell negative 10 end cell end table close parentheses plus t open parentheses table row cell negative 2 end cell row 3 end table close parentheses 

At time t equals 3 seconds the eagle is positioned at a point exactly 28 metres vertically above the mouse. 

The x comma space y and z directions are due east, due north and vertically upwards respectively with all distances in metres.

Write down a vector equation of a line the eagle should fly in if it is to take off at the time t equals 3 seconds and reach the ground at the same place as the mouse.

4b
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2 marks

During their subsequent motion, find the speed of

i)
the eagle,
ii)
the mouse.
4c
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3 marks

Luckily for the mouse, at the point where the eagle is exactly 5 metres from it, it finds a hole to hide in.

Find the amount of time the mouse had left before the eagle would have reached it.

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5a
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5 marks

A person is walking in a straight line and at a constant speed. The person moves from their initial point open parentheses 5 comma 1 close parentheses, through a second point open parentheses 2 comma negative 6 close parentheses and continues walking in a straight line. At the same time a leaf located at open parentheses 1 comma negative 5 comma 14 close parentheses falls from its starting position in a tree and moves with constant velocity in the direction of the vector a straight i plus open parentheses a minus 2 close parentheses straight j minus 4.92 straight k.   

The x comma space y and z directions are due east, due north and vertically upwards respectively with all distances in metres. 

Given that the person is 1.7 metres tall and that they are walking on a horizontal plane, find the coordinates of the point at which the leaf lands on the top of their head.

5b
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3 marks

The leaf takes 14.2 seconds to fall from its starting point in the tree to the point where it lands on the person’s head. Find the speed with which it falls.

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6a
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4 marks

The velocity of a tractor is given by v equals open parentheses table row cell 10 space sin space t end cell row cell negative 10 space cos space t end cell end table close parentheses. At the time, t seconds, the tractor is moving relative to an origin, O. At t equals 0 the position vector of the tractor is open parentheses table row cell negative 10 end cell row 0 end table close parentheses

Find

i)
the initial speed of the tractor
ii)
a position vector of the tractor at time, t , relative to the origin.
6b
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2 marks

Find the distance of the tractor from the origin at time, t.

6c
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2 marks

Describe the shape of the path that the tractor is making.

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7a
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6 marks

A ball is thrown from a height of 1.2 metres with initial velocity u equals open parentheses table row 4 row 15 end table close parentheses ms to the power of negative 1 end exponent. The ball moves freely under gravity. The velocity of the subsequent motion of the ball can be modelled by the vector

 v open parentheses t close parentheses equals open parentheses table row cell u subscript x end cell row cell u subscript y minus g t end cell end table close parentheses 

where u subscript xis the horizontal component of the initial velocity and u subscript yis the vertical component of the initial velocity. 

The ground below the point from which the ball was thrown can be considered the origin. It is assumed that any effects of air resistance will be negligible, the ball is modelled as a particle and g equals 9.81 space ms to the power of negative 2 end exponent.

Find

(i)
the displacement vector of the ball at the time, t ,
(ii)
the maximum height of the ball,
(iii)
the angle of elevation of the ball as it was thrown.
7b
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3 marks

Assuming that the ground is horizontal, find how far from the origin the ball is when it lands.

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8a
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5 marks

A whale is moving through the sea such that its position, at time t seconds, is given by the vector

 r equals open parentheses table row cell 1 third t squared minus 1 half t end cell row cell 1 half t squared plus 4 t end cell end table close parentheses 

Find

(i)
the initial speed of the whale,
(ii)
the speed of the whale after 10 seconds,
(iii)
the acceleration vector of the whale
8b
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4 marks

A turtle is riding on the back of the whale and at the time t equals 10 spaceseconds begins to swim away from it with velocity vector v equals 1 half T squared straight i plus 3 over 4 T squared straight j  , where T equals t minus 10 seconds. The whale continues to swim along the same path. 

Find the distance between the whale and the turtle 10 seconds after they separated.

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1a
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2 marks

Two ships straight A and straight B are travelling so that their position relative to a fixed point straight O at time t, in hours, can be defined by the position vectors bold italic r subscript bold A equals left parenthesis 2 minus t right parenthesis bold italic i plus left parenthesis 4 plus 3 t right parenthesis bold italic j and bold italic r subscript bold B equals left parenthesis t minus 8 right parenthesis bold italic i plus left parenthesis 29 minus 2 t right parenthesis bold italic j bold. 

The unit vectors i and space j are a displacement of 1 km due East and North of straight O respectively. 

Find the coordinates of the initial position of the two ships.

1b
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3 marks
Show that the two ships will collide and find the time at which this will occur.
1c
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2 marks
Find the coordinates of the point of collision.

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2a
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3 marks

A car, moving at constant speed, takes 4 minutes to drive in a straight line from point straight A open parentheses negative 3 comma space 5 close parentheses to point straight B open parentheses 7 comma space 11 close parentheses

At time t, in minutes, the position vector of the car relative to the origin can be given in the formbold space bold italic p equals a plus t b.

Find the vectors a and b.
2b
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3 marks

A cat has decided to take a nap at point straight X open parentheses 4 comma space 9 close parentheses. Show that the cat does not lie on the route along which the car drives.

2c
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6 marks
Find the shortest distance between the car and the cat during the movement of the car.

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3a
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2 marks

A bird takes off from a perch and flies at a constant speed in a straight line. The position of the bird in flight relative to its nest, (east, north and above/below the nest), can be described by the vector equation

r subscript 1 equals open parentheses table row 18 row 4 row cell negative 2 end cell end table close parentheses plus t open parentheses table row 272 row cell negative 360 end cell row 225 end table close parentheses.

 

All displacements are given in metres and t is the time in minutes.

Find the distance between the perch that the bird took flight from and its nest.

3b
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3 marks

Find the speed at which the bird is travelling. Give your answer in text kmh end text to the power of negative 1 end exponent.

3c
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4 marks

A second bird takes off at the same time as the first bird from a different perch and also flies in a straight line at a constant speed. The flight of the second bird, relative to the same nest, can be described by the vector equation

r subscript 2 equals open parentheses table row 12 row cell negative 8 end cell row cell negative 3 end cell end table close parentheses plus t open parentheses table row cell negative 187 end cell row cell negative 438 end cell row 80 end table close parentheses

Find the distance between the two birds after 8 minutes of flying.

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4a
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2 marks

A drone travels in a straight line and at a constant speed. It moves from an initial point straight A left parenthesis 4 comma 5 comma negative 2 right parenthesis comma through a second point straight B left parenthesis 7 comma negative 1 comma 0 right parenthesis and continues moving along the same line. The person controlling the drone is located at straight C left parenthesis 2 comma 3 comma 2 right parenthesis..

The x comma y spaceand z directions are due east, due north and vertically upwards respectively with all distances in metres.

Write down an equation for the line along which the drone travels.

4b
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3 marks

At some point straight P on its flight, the drone is at the same vertical height as the person controlling the drone.

Find the coordinates of point straight P.

4c
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2 marks

Find the distance between straight P and the person controlling the drone.

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5a
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1 mark

Two snails are taking part in a snail race starting from the same point and moving in a straight line. The position of the first snail S subscript 1 is given by the equation

 r equals open parentheses table row 5 row 1 end table close parentheses plus t open parentheses table row 1 row cell negative 2 end cell end table close parentheses

 

The displacement of the second snail S subscript 2, relative to the finish point, is given by

s left parenthesis t right parenthesis equals 8 minus 3 t squared.

 All distances are in centimetres and time is in minutes.

 Write down the distance that the snails race.

5b
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2 marks

Find an expression for the velocity of S subscript 2 at time t.

5c
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5 marks

Find the displacement of S subscript 2 from the finishing point when the speed of the two snails is equal.

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6a
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2 marks

A ball is pushed off the top of a 150 m tall skyscraper with an initial velocity of open parentheses table row cell 1.5 end cell row 0 end table close parentheses ms to the power of negative 1 end exponent.

The point at which the ball is pushed can be considered the origin of a Cartesian coordinate system. It is assumed that any effects of air resistance will be negligible and g equals 9.81 ms to the power of negative 2 end exponent.

Find the velocity vector at time t.

6b
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2 marks

displacement vector of the ball at time t.

6c
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2 marks

Find the time at which the ball reaches the ground.

6d
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2 marks

Find the total horizontal distance travelled by the ball.

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7a
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4 marks

Two aeroplanes are observed flying in straight lines, with respect to the airport located at left parenthesis 0 comma 0 comma 0 right parenthesis. The flightpaths l subscript A and l subscript B, of aeroplanes A and B respectively, can be defined by:

 l subscript A colon r equals open parentheses table row 6 row 3 row cell negative 2 end cell end table close parentheses plus alpha open parentheses table row cell negative 1 end cell row cell negative 4 end cell row 3 end table close parentheses

l subscript B colon s equals open parentheses table row cell negative 7 end cell row cell negative 1 end cell row 5 end table close parentheses plus beta open parentheses table row 1 row cell negative 2 end cell row 1 end table close parentheses

where alpha and beta is the time elapsed in minutes since the start of the observation for each aeroplane. All distances are in kilometres.

The flightpaths intersect at point straight P.

Find the values of alpha and beta and hence show that the two planes will not collide.

7b
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4 marks

Find

(i)
the coordinates of the point at which the flightpaths intersect,

 

(ii)
the distance between the airport and the point P.

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8a
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2 marks

A particle starts from a position at left parenthesis 0 comma 0 right parenthesis space and moves such that its velocity at time t, in seconds, is given by v equals open parentheses table row cell 2 e to the power of 3 t end exponent end cell row cell e to the power of 3 t end exponent minus 4 end cell end table close parentheses. All distances are in metres.

Find an expression for the acceleration of the particle at time t.

 

8b
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4 marks

Find an expression for the position of the particle at time t.

8c
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3 marks

Find the value of t such that the speed of the particle is 6 space ms to the power of negative 1 end exponent.

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1a
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2 marks

Two drones X and Y are being flown over an area of rainforest to look for signs of illegal logging. Their positions relative to the observation centre, are given by

r subscript x equals open parentheses table row cell negative 3 end cell row cell 1.6 end cell row cell 2.5 end cell end table close parentheses plus t open parentheses table row 2 row cell negative 2 end cell row 1 end table close parentheses  and r subscript y equals open parentheses table row cell 2.5 end cell row 0 row cell negative 2 end cell end table close parentheses plus t open parentheses table row cell 1.5 end cell row 6 row 4 end table close parentheses

 

at time t  minutes after take-off, 0 less or equal than t less than 20. All distances are in metres.

Verify that the two drones will not collide.
1b
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6 marks
Find the shortest distance between the two drones and the time at which it occurs.
1c
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6 marks

A third drone Z begins its flight at t equals 8 and its position relative to the observation centre is given by r subscript z equals open parentheses table row 2 row cell 1.5 end cell row cell 4.5 end cell end table close parentheses plus t open parentheses table row 3 row 4 row 1 end table close parentheses 

Each drone can observe a circular area of ground,  A comma such that A equals 1.8 h squared where h is the height of the drone above the ground in metres.

Show that the area of ground that can be observed by drone Z five minutes after it takes off overlaps with the area of ground that can be observed by drone Y at that time.

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2a
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6 marks

A car is moving at a constant speed of 15 ms-1 in the direction parallel to the vector 3 straight i minus 6 straight j.  Two birds are perched at points straight A left parenthesis 17 comma space 28 comma space 16 right parenthesis  and straight B open parentheses negative 48 comma space 128 comma space 26 close parentheses. 

At t equals 0, the car is located at open parentheses 2 comma space 4 comma space 0 close parentheses  and the bird at point A starts to fly at a constant velocity of  fraction numerator 7 square root of 365 over denominator 10 end fraction ms-1. The bird at point B begins to fly at a constant velocity in the direction of the vector 52 straight i minus 60 straight j minus 9 k when t equals 1.2. 

When bird A reaches the position of open parentheses 44 comma negative 24 comma space 4 close parentheses, both birds and the car lie in a straight line.

Find the equation of the line along which the birds and car lie.
2b
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6 marks
Find the speed at which bird B is travelling.

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3
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11 marks

Consider the following diagram depicting imaginary lines connecting five points in space:

mi_q3_3-9_modelling-with-vectors_very-hard_ib_ai_hl_maths_dig

Points A comma space B comma space C and D are the locations, respectively, of the stars Arccirclus, Betacarotjuse, α-Capella and Denomineb.  Point S is the location of the Stellamortis battle station, a planet-killing atrocity being built by the evil Galactic Imperium.  Coordinates are given relative to an origin point in accordance with the standard x comma space y comma space z coordinate system, and the units for all coordinates are parsecs. 

The forces of the Star Rebellion are prepared to launch a strike to destroy the battle station, but they are unsure of its exact location.  According to data recovered from a smuggled droid, however, the following facts are known about the location of point S : 

  • Point S is in the First Octant of the galaxy, where x comma space y space and space z coordinates are all positive. 
  • The distance from point C  to point S is exactly  45 square root of 2 spaceparsecs. 
  • Points B comma space C comma space D and S  form the base of a pyramid, with its apex at point A.
  • The point on BD closest to point A is also the point where the two diagonals of the pyramid’s base intersect.

As the rebellion’s Chief Mathematician, it is your job to use the information provided to find the exact coordinates of point S.  The fate of the galaxy is in your mathematical hands!

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4a
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8 marks

An oyster on the edge of a coral reef projects a microbubble into a jet stream and its subsequent motion can be modelled as a position vector. The microbubble reaches a maximum height and then moves back downwards in front of the oyster and continues down into the sea below.

The acceleration of the microbubble can be modelled by the vector

 a equals open parentheses 0.4 straight i minus 0.6 t straight j close parentheses straight m space straight s to the power of negative 2 end exponent 

Taking the origin to be the point at which the oyster is sitting, the unit vectors straight i and straight j are a displacement of 1 m along the horizontal and vertical axis of a Cartesian coordinate system respectively.

Given that it takes 5 seconds until the microbubble is at the same horizontal height as the oyster again, and that the horizontal distance of the microbubble from the oyster at t equals 5 is double that of when it is at its maximum height, find
(i)
the maximum height above the oyster that the microbubble reaches,
(ii)
the position vector of the microbubble at time, t .
4b
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4 marks
The seabed is 22 metres below the level of the oyster. Find the speed of the microbubble at the moment when it hits the seabed.

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5a
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4 marks

A boat is moving such that its position vector when viewed from above at time t  seconds can be modelled by

 r equals open parentheses table row cell 10 minus a space sin open parentheses πt over 600 close parentheses end cell row cell b open parentheses 1 minus cos open parentheses πt over 600 close parentheses close parentheses end cell end table close parentheses 

with respect to a rectangular coordinate system from a point O, where the non-zero constants a  and b can be determined. All distances are given in metres. 

The boat leaves its mooring point at time t equals 0 seconds and 5 minutes later is at the point with coordinates open parentheses negative 20 comma space 40 close parentheses

Find

(i)
the values of a and b
(ii)
the displacement of the boat from its mooring point.
5b
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2 marks
Find the velocity vector of the boat at time t seconds.
5c
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3 marks

Find the time that the boat returns to its mooring point and the acceleration of the boat at this moment.

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6a
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3 marks

A small stunt plane is heading in to land at an airport with acceleration given by the vector

 a equals open parentheses table row cell negative 0.06 end cell row cell negative 0.03 t end cell end table close parentheses space ms to the power of negative 2 end exponent 

The straight i component represents horizontal motion and the straight j component represents vertical motion. The start of the runway is considered the origin and the runway runs along the
­horizontal axis. When t equals 10 seconds the velocity of the plane is 0.9 straight i minus 3.5 straight j space ms to the power of negative 1 end exponent and the plane is 27 metres vertically above the start of the runway.  

Find

(i)
the time in seconds at which the stunt plane lands on the runway,
(ii)
the distance of the stunt plane from the start of the runway when it lands.
6b
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2 marks

Find the speed with which the stunt plane lands.

6c
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1 mark

At the moment of landing, this particular type of stunt plane needs to have a deceleration of between 0.4 and 0.5 ms-2.

Decide whether the stunt plane has landed within the safe landing limits.

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7a
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3 marks

Two children are observing the movement of some worms in their garden. The worms are placed on the ground at the same time and begin to move instantly. The first worm, W subscript 1 moves with velocity at time t seconds given by the equation

v equals open parentheses table row cell e to the power of negative 0.4 t end exponent open parentheses a space cos space t minus b sin space t close parentheses end cell row cell e to the power of negative 0.4 t end exponent open parentheses a space sin space t plus b cos space t close parentheses end cell end table close parentheses. 

The second worm, W subscript 2 has position vector given by

 r equals open parentheses table row cell 4 e to the power of negative 0.5 t end exponent space cos space t end cell row cell 3 e to the power of negative 0.2 t end exponent space sin space t end cell end table close parentheses

All distances are in metres and time is in seconds.

Find the velocity vector of W subscript 2 spaceat time t seconds.

 

7b
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5 marks

Given that both worms are travelling parallel to each other in the same direction and at the same speed at time t equals 20, find

(i)
the values of a and b,
(ii)
the speed at which the two worms are travelling at this moment.

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8a
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4 marks

A spider starts from the origin and begins to weave a web such that her velocity vector at time t  seconds with respect to a rectangular coordinate system can be modelled by

 v equals open parentheses table row cell a space sin open parentheses b t close parentheses plus sin open parentheses t close parentheses end cell row cell cos open parentheses t close parentheses minus a space cos open parentheses b t close parentheses end cell end table close parentheses 

where a less than 0 and 0 less than b less than 0.15.

Find an expression for the position vector of the spider at time t, in terms of a and b.

8b
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4 marks

Given that at time t equals 9 straight pi seconds the spider is moving parallel to the y-axis with a speed of  7 over 3 ms to the power of negative 1 end exponent , find the values of a and b.

8c
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1 mark

Find the earliest time at which the spider is weaving its web parallel to the x-axis.

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