Kinematics (DP IB Maths: AI HL)

Exam Questions

5 hours29 questions
1a
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2 marks

A skydiver jumps from a moving aircraft at a point directly above a fixed point, O, on the ground.  The trajectory of the skydiver is then modelled by the function

h left parenthesis x right parenthesis equals 3200 minus 0.5 x squared

where h straight m is the height of the skydiver above the ground and x spacestraight m is the horizontal distance along the ground from point O.           

(i)       Explain the significance of the value 3200 in the model.

(ii)      Calculate the horizontal distance the skydiver covered upon landing.



1b
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2 marks

Sketch a graph of h against x.

1c
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1 mark

Explain why the model is not suitable for values of x larger than 80 m.

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2a
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2 marks

A particle moves along a horizontal line starting at the point O. The displacement-time graph for the first 20 seconds of its motion is shown below. Displacement is measured in metres.

ib2a-ai-sl-5-5-ib-maths-medium

(i)      Write down the displacement of the particle after 2 seconds.

(ii)     Write down the displacement of the particle after 4 seconds.

2b
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1 mark

Find the velocity of the particle between 13 and 20 seconds.

2c
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1 mark

Find the speed of the particle between 7 and 10 seconds.

2d
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2 marks

Find the total distance travelled by the particle after 20 seconds.

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3a
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2 marks

A cricket ball is projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

h left parenthesis t right parenthesis equals 13 t minus 4.9 t squared space space space space space space t greater than 0

where h metres is the height of the cricket ball above ground level after t seconds.

Find the times at which the cricket ball is exactly 3 m above the ground.

3b
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1 mark

For how long is the cricket ball at least 3 m above the ground?

3c
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2 marks

A player catches the cricket ball (on its way down) at a height of 0.8 m above the ground.

Find the length of time the ball was in the air.

3d
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2 marks

Find the total distance travelled by the ball.

3e
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2 marks

Find the velocity of the cricket ball at t equals 1 second.

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4a
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1 mark

A softball is thrown upwards from the top of a 10 m tall building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals H plus 7.8 t minus 4.9 t squared space space space space space space space space t greater than 0

Write down the value of H.

4b
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2 marks

Find the height of the ball after 2 seconds.

4c
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2 marks

Find the time at which the ball is at the same height as it was when thrown.

4d
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2 marks

Find the time the ball first hits the ground.

4e
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3 marks

Find h to the power of apostrophe apostrophe left parenthesis t right parenthesis and hence show that the acceleration at any time is negative 9.8 straight m divided by straight s squared.

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5a
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2 marks

A particle moves along a straight line with a velocity,  straight v space ms to the power of negative 1 end exponent comma given by  v equals 2 to the power of t minus 2   where t  is measured in seconds such that  0 less or equal than t less or equal than 4.

Find the acceleration of the particle at time t equals 2 .

5b
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1 mark

State the time when the particle comes to rest.

5c
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3 marks

Find the total distance travelled by the particle.

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6a
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4 marks

A particle is found to have an acceleration,a space ms to the power of negative 2 end exponent, according to the function

               a equals 1 over t squared plus sin space t,where t greater or equal than 1

Find an expression for the velocity,v , of the particle given that  v left parenthesis 1 right parenthesis equals 1 . 

6b
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2 marks

Find the velocity of the particle at space t equals 2.

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7a
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2 marks

A particle, moving in a straight line, is found to have a velocity space v equals sin space t plus cos space 2 t where v is measured in ms to the power of negative 1 end exponent and  time t is measured in seconds such that 0 less or equal than t less or equal than 5.

Find the time(s) when the particle is instantaneously at rest.

7b
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1 mark

Find the time(s) when the particle changes direction.

7c
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3 marks

Find the distance travelled in the first second of motion.

7d
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3 marks

Find the acceleration of the particle at the instant it first changes direction.

7e
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4 marks

Find the displacement of the particle from its starting point to the point when space t equals 5.

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8a
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4 marks

A particle is moving along a straight line.  The position of the particle at time t seconds, measured in metres relative to a fixed origin point, is denoted by  x left parenthesis t right parenthesis.

The particle starts at the origin at time t equals 0 ,  and its motion over the next eight seconds is described by the equation

x left parenthesis t right parenthesis equals fraction numerator 1 over denominator cos squared open parentheses straight pi over 20 t close parentheses end fraction minus 3 comma space space space space space space space 0 less or equal than t less or equal than 8

Find an expression for x left parenthesis t right parenthesis.

8b
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3 marks

Hence determine the maximum distance of the particle from the origin during the first eight seconds of its movement.

8c
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2 marks

Find the change in displacement of the particle during the first eight seconds of its movement.

8d
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2 marks

Find the total distance travelled by the particle during the first eight seconds of its movement.

8e
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3 marks

Find an expression for the particle’s acceleration stack x space with ¨ on top open parentheses t close parentheses.

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9a
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3 marks

A particle is moving along a straight line.  The position of the particle at any given time, measured in metres relative to a fixed origin point, is denoted by x.

It is known that the velocity,straight v space ms to the power of negative 1 end exponent , of the particle is dependent on the particle’s position, and that the velocity may be described by the equation

v left parenthesis x right parenthesis equals square root of 1 minus x squared end root comma space space space space space space space space space minus 1 less or equal than x less or equal than 1

Use the chain rule to explain why the acceleration, a space ms to the power of negative 2 end exponent, of the particle may be expressed in the form 

a equals v fraction numerator space d v over denominator d x end fraction

9b
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4 marks

Show that the derivative of  square root of 1 minus x squared end root  is negative fraction numerator x over denominator square root of 1 minus x squared end root end fraction .

9c
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2 marks

Hence find an expression for the acceleration of the particle in terms of x, being sure to indicate the domain of x values for which the expression is valid.

9d
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3 marks

Identify the minimum and maximum values of

          (i)     the speed of the particle

          (ii)    the magnitude of the particle’s acceleration

along with the values of x for which those occur.

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1a
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2 marks

A golf ball is hit from a point O on a horizontal golf course, and travels at all times in a vertical plane that passes through O.  The trajectory of the golf ball is modelled by the equation

y equals negative x squared over 180 plus x

where x  and y  are respectively the horizontal and vertical displacements, in metres, of the golf ball relative to point O.  Upwards is taken to be the positive direction for the vertical displacement.

Write down an appropriate domain for the model, explaining why the domain given is suitable within the context of the question.

1b
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2 marks

Sketch the graph of y against x , labelling any intersections with the coordinate axes.

1c
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1 mark

Find the maximum height reached by the golf ball.

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2a
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2 marks

A ball is placed at the midpoint, X, of a horizontal tube of length 120 cm.  A player is positioned at each end of the tube and is required to pump air into the tube in order to move the ball.

The velocity-time graph for a 20 second game is shown below, where v  is the velocity of the ball in cm   straight s to the power of negative 1 end exponent  and t  is the time in seconds since the start of the game.

ib2a-ai-sl-5-6-ib-maths-hard

Write down the maximum speed of the ball during the game and the time interval during which that maximum speed occurs.

2b
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3 marks

There are two periods of time during which the ball is decelerating. Find the magnitude of the deceleration that the ball undergoes after it has changed direction for the first time.

2c
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3 marks

Find the total distance that the ball travels in the game.

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3a
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1 mark

A particle moves along a straight line relative to a fixed point, P.  The motion of the particle can be modelled by the function

s left parenthesis t right parenthesis equals t cubed over 20 minus fraction numerator 17 t squared over denominator 8 end fraction plus 18 t minus 4 comma space space space space space space space space space space space space 0 less or equal than t less or equal than 30

where s  is the horizontal displacement in metres from point P and t  is the time in seconds.

Write down the initial distance of the particle from point P.

3b
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1 mark

Find the final displacement of the particle from P.

3c
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2 marks

Find an expression, in terms of t , for the velocity of the particle.

3d
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5 marks

Using your answer to part (c), find the times at which the particle is stationary.

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4a
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1 mark

A skier performs a jump in a competition.  She leaves the ground at point Q and travels in a vertical plane through Q, landing at point R.  This can be seen in the diagram below.

ib3a-ai-sl-5-6-ib-maths-hard

The trajectory of the jump can be modelled as

h left parenthesis x right parenthesis equals negative x over 80 open parentheses x minus 35 close parentheses plus 30 comma space space space space space space space space space 0 less or equal than x less than 70

where x  is the horizontal displacement of the skier from point Q and h  is the vertical displacement of the skier relative to point R.

Write down the vertical distance between points Q and R.

4b
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2 marks

Show that when the skier is again at the same vertical height from which she started, then her horizontal distance from point Q is 35 m.

4c
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6 marks

Find an expression for h to the power of apostrophe left parenthesis x right parenthesis  and hence find the total distance in the vertical direction that the skier will travel.  Show your working.

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5a
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2 marks

A particle moves in a straight line with a velocity v  ms-1 given by v left parenthesis t right parenthesis equals square root of open parentheses t squared plus 2 t close parentheses end root minus 3   , where t  is measured in seconds such that  0 less or equal than t less or equal than 5.

Find the acceleration of the particle at time t equals 3.7.

.

5b
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3 marks

(i)       Find the change in the particle’s displacement between the times  t equals 0  and  t equals 1 .

(ii)
Explain what this change in displacement tells you about the particle’s motion between those two times.
5c
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3 marks

Find the total distance travelled by the particle.

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6a
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1 mark

A marble is projected along a marble run that travels in a vertical plane through a fixed point O.  The marble’s vertical distance, h , in cm above point O can be modelled by

h left parenthesis t right parenthesis equals 15 minus 10 t plus 2 t squared comma space space space space space space space 0 less or equal than t less or equal than 6

where t  is the time in seconds after the marble is projected.

Write down the initial height of the marble relative to the fixed point O.

6b
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6 marks

Find the time at which the marble reaches its lowest point, and find the total vertical distance that the marble has travelled up to that time.

6c
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4 marks

Find the velocity and acceleration of the marble at the end of the given time period.

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7a
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4 marks

A particle is found to have a velocity, v  ms-1, that can be expressed by the function

 v equals t cubed c o s space t comma space space space space t greater or equal than 0

Find an expression for the acceleration, a, of the particle

7b
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2 marks

Hence find the acceleration of the particle at the time t equals 5.2 .

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8a
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1 mark

A particle moves along a straight line such that its displacement s in metres from a fixed point O is given by

s left parenthesis t right parenthesis equals 1 half t minus sin space 2 t comma space space space space space space space space space for space space 0 less or equal than t less or equal than 6

where t is the time in seconds.

Write down the number of changes of direction that the particle makes.

 

8b
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3 marks

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

8c
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3 marks

Find the maximum velocity and the time(s) at which it occurs.

8d
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3 marks

Considering the total distance travelled by the particle, calculate the percentage of that total distance that the particle travels in the first 2 seconds of its movement.

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9a
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7 marks

A particle is moving along a straight line.  The position of the particle at time  seconds, measured in metres relative to a fixed origin point, is denoted by x open parentheses t close parentheses .

The particle starts at rest at the origin at time t equals 0 ,  and its motion over the next six seconds is described by the equation

x with ¨ on top left parenthesis t right parenthesis equals fraction numerator 4 pi squared over denominator 9 end fraction sin space open parentheses fraction numerator 2 pi t over denominator 3 end fraction close parentheses minus straight pi squared over 36 space cos space open parentheses fraction numerator pi t over denominator 12 end fraction close parentheses comma space space space space space space space space 0 less or equal than t less or equal than 6

Find expressions for

(i)       x with dot on top left parenthesis t right parenthesis

(ii)      x left parenthesis t right parenthesis.

9b
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4 marks

Find the maximum value that each of the following quantities takes on during the first six seconds of movement, as well as the time t at which those maximum values occur:

(i)      the distance of the particle from the origin

(ii)     the speed of the particle

(iii)    the magnitude of the particle’s acceleration.

9c
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2 marks

Find the total distance travelled by the particle during the first six seconds of its movement.

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10a
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3 marks

A particle is moving along a straight line.  The position of the particle at any given time, measured in metres relative to a fixed origin point, is denoted by x.

It is known that the velocity, v space ms to the power of negative 1 end exponent, of the particle is dependent on the particle’s position, and that the velocity may be described by the function

v left parenthesis x right parenthesis equals negative square root of 4 minus 9 x squared end root comma space space space space space space space space space minus 2 over 3 less or equal than x less or equal than 2 over 3

Show that the acceleration, a space ms to the power of negative 2 end exponent, of the particle may be expressed in the form

a equals v space fraction numerator d v over denominator d x end fraction

10b
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1 mark

Hence find a function giving the acceleration of the particle in terms of x.

10c
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3 marks

Identify the minimum and maximum values of

          (i)     the speed of the particle

          (ii)     the magnitude of the particle’s acceleration

along with the values of x for which those occur.

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1a
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3 marks

A toy car starts from a fixed point P and moves along a horizontal race track.  The horizontal displacement, s  cm, of the toy car from point P can be modelled by the function

s left parenthesis t right parenthesis equals 1 over 1000 t left parenthesis t squared minus 190 t plus 8400 right parenthesis comma space space space space space space space space space 0 less or equal than t less or equal than 140

where t  is the time in seconds since leaving point P.

Sketch a graph of s left parenthesis t right parenthesis  against t .

1b
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4 marks

Find an expression for a open parentheses t close parentheses and hence find the acceleration of the toy car at t equals 85 spaceseconds.

1c
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3 marks

Find the greatest speed that the toy car reaches when travelling such that its displacement is decreasing.

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2a
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2 marks

The level of water, h m, in an estuary relative to the mean sea level, is observed over a  24-hour period starting from midnight and is modelled by the function

h left parenthesis t right parenthesis equals sin left parenthesis 0.262 t minus 0.5 right parenthesis minus 3 space cos left parenthesis 0.524 t right parenthesis plus 1

where t  is the time in hours after midnight.

Find the rate of change of the height of the water level at 7 am.

2b
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4 marks

Find the percentage of time within the 24-hour period that the water level remains above the mean sea level.

2c
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4 marks

The scientists observing the water level in the estuary anchor a buoy in place such that its horizontal position is fixed, but it is able to move up and down vertically with the changing water level

Find the total vertical distance that the buoy moves during the 24-hour time period.

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3a
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3 marks

For a particle P moving in a straight line let  fraction numerator space d v over denominator d t end fraction equals left parenthesis sin left parenthesis t right parenthesis space right parenthesis left parenthesis cos left parenthesis 4 t right parenthesis space right parenthesis  for  0 less or equal than t less or equal than 2 ,  where v is the velocity of the particle and t is the elapsed time in seconds.

Sketch the graph of   fraction numerator d v over denominator d t end fraction on the grid below.

ib3a-ai-sl-5-6-ib-maths-veryhard

3b
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5 marks

Find the times at which the points of inflection would occur on a displacement-time graph representing the particle’s movement, and explain the significance of these points in the context of this question.

3c
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2 marks

Hence find the values of t for which the displacement-time graph would be concave down.

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4a
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2 marks

Two particles, straight P subscript 1  and straight P subscript 2 ,  are observed moving along a straight line.  The displacements of the particles, respectively s subscript 1  and s subscript 2 , in metres relative to a fixed point O can be modelled for 0 less or equal than t less or equal than 3  by the following functions

s subscript 1 left parenthesis t right parenthesis equals 1 half sin open parentheses t minus 0.9 close parentheses minus cos left parenthesis 2 t minus 1.8 right parenthesis minus 1

s subscript 2 left parenthesis t right parenthesis equals cos left parenthesis 6 t minus 5.4 right parenthesis minus sin left parenthesis t minus 0.9 right parenthesis plus 2.5

where t  is the time in seconds from the start of the observation.

Find an expression for the distance between the two particles at time t.

4b
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3 marks

Hence find

(i)       the maximum distance of the particles from one another

(ii)
the time at which the maximum distance between the particles occurs.
4c
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3 marks

A collision occurs between the particles during the time of observation.

Find the velocity of each of the particles 0.5 seconds before the time that they collide.

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5a
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2 marks

A particle starts from point X and moves in a straight line. The graph below shows its velocity,v ms-1 after t  seconds for 0 less or equal than t less or equal than 8 .

ib5a-ai-sl-5-6-ib-maths-veryhard

The particle has an instantaneous velocity of 0  ms-1 at t equals 0 comma t equals 3 comma t equals 5  and t equals 8  .

The function space s left parenthesis t right parenthesis represents the displacement of the particle from point X after seconds.

It is known that the particle travels 22 metres in the first 3 seconds.

It is also known that s left parenthesis 3 right parenthesis equals s left parenthesis 7 right parenthesis  and integral subscript 3 superscript 5 v space dt equals 9 .

Find the value of s left parenthesis 5 right parenthesis minus s left parenthesis 3 right parenthesis .

5b
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7 marks

Find the total distance travelled by the particle in the first 7 seconds.

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6a
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1 mark

A particle P moves along a straight line.  The velocity of the particle after t  seconds, v subscript P ms to the power of negative 1 end exponent  is given by

v subscript P equals t squared cos open parentheses straight pi over 4 t close parentheses comma space space space space space space space space space 0 less or equal than t less or equal than 10

Write down the first value of t at which P changes its direction of motion.

6b
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3 marks

Find the total distance travelled by P during the periods when its speed is increasing.

6c
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4 marks

A second particle, Q, also moves along a straight line.  Its velocity after t  seconds, v subscript Q ms to the power of negative 1 end exponent is given by 

v subscript Q equals 6 t plus 2 comma space space space space space space space space space space space space space space 0 less or equal than t less or equal than 8

After k  seconds, the total distance that Q has travelled is the same as the distance that P travels during its periods of increasing speed.

Find the value of k .

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7a
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2 marks

A particle moves in a straight line starting from point P.  The particle is found to have a velocity, v   ms-1, given by the piecewise function

v open parentheses t close parentheses equals open curly brackets table row cell 9 t minus 3 t squared end cell cell 0 less or equal than t less or equal than 4 end cell row cell negative 3 t plus 16 over t cubed minus 1 fourth end cell cell 4 less than t less or equal than 10 end cell end table close

Find the maximum velocity reached by the particle and the time at which that maximum velocity is reached.

7b
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5 marks

Find an expression for the displacement of the particle from the starting point P at time t , given that the displacement of the particle from point P at the end of the time period is   minus 119.08  m.

7c
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4 marks

Find the total distance travelled by the particle.

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8a
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2 marks

A particle A moves along a horizontal straight line L subscript 1 .  The displacement, s subscript A  cm, of particle A from a fixed point P on L subscript 1  is given by the function

s subscript A left parenthesis t right parenthesis equals 1 half t minus 2 t cubed e to the power of negative 0.3 t end exponent plus 24 comma space space space space space space space space 0 less or equal than t less or equal than 22

where t  is the time in seconds from the start of the motion.

Starting at the same time, another particle, B, moves along a horizontal straight line L subscript 2  which is parallel to L subscript 1 .

The velocity of particle B, straight v subscript straight B cm   straight s to the power of negative 1 end exponent , at time t  seconds is given by

v subscript B left parenthesis t right parenthesis equals 1 half t plus 12 comma space space space space space space space 0 less or equal than t less or equal than 22

Find the value(s) of for which particle A is at point P.

8b
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2 marks

Find the value of t at which particle A first changes direction.

8c
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3 marks

Find the total distance travelled by particle A in the first 8 seconds of its motion.

8d
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7 marks

The displacement, s subscript B  cm, of particle B is measured relative to a fixed point Q on L subscript 2 .

Given that s subscript A left parenthesis 0 right parenthesis equals s subscript B left parenthesis 5 right parenthesis , find:

(i)

the displacement function s subscript B  for particle B

(ii)
the displacement of each particle at the time when the displacement of particle A from point P is the same as the displacement of particle B from point Q.

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9a
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7 marks

A particle is moving along a straight line.  The position of the particle at time t seconds, measured in metres relative to a fixed origin point, is denoted by x open parentheses t close parentheses .

The particle starts at the origin at time  t equals 0  with a velocity of negative 3 space ms to the power of negative 1 end exponent ,  and its motion over the next ten seconds is described by the equation

x with ¨ on top left parenthesis t right parenthesis equals 3 over 32 e to the power of fraction numerator 3 t over denominator 8 end fraction end exponent minus fraction numerator 9 pi squared over denominator 25 end fraction c o s fraction numerator 3 pi t over denominator 5 end fraction comma space space space space space 0 less or equal than t less or equal than 10

Considering the total distance travelled by the particle over the ten seconds, calculate the percentage of that total distance that the particle travels in the first five seconds of its movement.

9b
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5 marks

Find the greatest distance from the origin point that the particle reaches, and the time t  at which that greatest distance is reached.

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10a
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2 marks

Professor Goodwin Vundera, a kinematics researcher, has been studying a particular type of particle.  The particle is known only to move along a straight line, with its acceleration,a space straight m   straight s to the power of negative 2 end exponent , and velocity,v space ms to the power of negative 1 end exponent , both dependent on the particle’s displacement, x m, with reference to a fixed origin point.

The professor has defined a new real-valued function, the ‘Vundera function’, which he believes captures all necessary information about the motion of the particle.  This Vundera function, W, is defined by  W left parenthesis x right parenthesis equals fraction numerator a left parenthesis x right parenthesis over denominator v open parentheses x close parentheses end fraction ,  where  a open parentheses x close parentheses and v open parentheses x close parentheses are functions describing, respectively, the particle’s acceleration and velocity in terms of x.  For one of the particles studied by the professor, the associated Vundera function is found to be

W left parenthesis x right parenthesis equals fraction numerator 2 x over denominator square root of 225 minus 9 x squared end root end fraction comma space space space space space space space a less than x less than b

where a and b are real constants.

Given that W has the largest possible domain, write down the values of a and b.

10b
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6 marks

Additionally it is known that when x equals 4, the velocity of the particle is negative 2 space ms to the power of negative 1 end exponent.

By using the above information and solving an appropriate indefinite integral, find expressions for the functions v open parentheses x close parentheses and a open parentheses x close parentheses associated with the particle.

10c
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3 marks

Explain the relationship between the particle’s speed and acceleration as x varies across all the values in the domain.

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